Normalized defining polynomial
\( x^{45} - 12 x^{44} - 80 x^{43} + 1424 x^{42} + 1516 x^{41} - 74554 x^{40} + 63158 x^{39} + \cdots + 1507921 \)
Invariants
| Degree: | $45$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[45, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(252\!\cdots\!929\)
\(\medspace = 13^{30}\cdot 61^{42}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(256.41\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $13^{2/3}61^{14/15}\approx 256.41105616096826$ | ||
| Ramified primes: |
\(13\), \(61\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $45$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(793=13\cdot 61\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{793}(256,·)$, $\chi_{793}(1,·)$, $\chi_{793}(386,·)$, $\chi_{793}(131,·)$, $\chi_{793}(391,·)$, $\chi_{793}(9,·)$, $\chi_{793}(269,·)$, $\chi_{793}(16,·)$, $\chi_{793}(789,·)$, $\chi_{793}(22,·)$, $\chi_{793}(217,·)$, $\chi_{793}(666,·)$, $\chi_{793}(672,·)$, $\chi_{793}(625,·)$, $\chi_{793}(42,·)$, $\chi_{793}(300,·)$, $\chi_{793}(562,·)$, $\chi_{793}(302,·)$, $\chi_{793}(178,·)$, $\chi_{793}(443,·)$, $\chi_{793}(705,·)$, $\chi_{793}(196,·)$, $\chi_{793}(198,·)$, $\chi_{793}(74,·)$, $\chi_{793}(718,·)$, $\chi_{793}(81,·)$, $\chi_{793}(339,·)$, $\chi_{793}(469,·)$, $\chi_{793}(729,·)$, $\chi_{793}(347,·)$, $\chi_{793}(607,·)$, $\chi_{793}(352,·)$, $\chi_{793}(144,·)$, $\chi_{793}(484,·)$, $\chi_{793}(230,·)$, $\chi_{793}(321,·)$, $\chi_{793}(744,·)$, $\chi_{793}(367,·)$, $\chi_{793}(497,·)$, $\chi_{793}(757,·)$, $\chi_{793}(118,·)$, $\chi_{793}(503,·)$, $\chi_{793}(378,·)$, $\chi_{793}(508,·)$, $\chi_{793}(510,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{11}a^{27}+\frac{3}{11}a^{26}+\frac{3}{11}a^{25}+\frac{2}{11}a^{24}-\frac{3}{11}a^{23}-\frac{4}{11}a^{22}+\frac{4}{11}a^{21}+\frac{5}{11}a^{20}+\frac{2}{11}a^{19}+\frac{5}{11}a^{18}-\frac{4}{11}a^{17}+\frac{3}{11}a^{16}+\frac{5}{11}a^{15}+\frac{5}{11}a^{14}-\frac{2}{11}a^{12}+\frac{5}{11}a^{11}+\frac{3}{11}a^{10}+\frac{2}{11}a^{8}-\frac{3}{11}a^{7}-\frac{2}{11}a^{6}-\frac{4}{11}a^{5}-\frac{1}{11}a^{4}-\frac{4}{11}a^{3}-\frac{3}{11}a^{2}-\frac{4}{11}a+\frac{2}{11}$, $\frac{1}{11}a^{28}+\frac{5}{11}a^{26}+\frac{4}{11}a^{25}+\frac{2}{11}a^{24}+\frac{5}{11}a^{23}+\frac{5}{11}a^{22}+\frac{4}{11}a^{21}-\frac{2}{11}a^{20}-\frac{1}{11}a^{19}+\frac{3}{11}a^{18}+\frac{4}{11}a^{17}-\frac{4}{11}a^{16}+\frac{1}{11}a^{15}-\frac{4}{11}a^{14}-\frac{2}{11}a^{13}-\frac{1}{11}a^{11}+\frac{2}{11}a^{10}+\frac{2}{11}a^{9}+\frac{2}{11}a^{8}-\frac{4}{11}a^{7}+\frac{2}{11}a^{6}-\frac{1}{11}a^{4}-\frac{2}{11}a^{3}+\frac{5}{11}a^{2}+\frac{3}{11}a+\frac{5}{11}$, $\frac{1}{11}a^{29}-\frac{2}{11}a^{25}-\frac{5}{11}a^{24}-\frac{2}{11}a^{23}+\frac{2}{11}a^{22}-\frac{4}{11}a^{20}+\frac{4}{11}a^{19}+\frac{1}{11}a^{18}+\frac{5}{11}a^{17}-\frac{3}{11}a^{16}+\frac{4}{11}a^{15}-\frac{5}{11}a^{14}-\frac{2}{11}a^{12}-\frac{1}{11}a^{11}-\frac{2}{11}a^{10}+\frac{2}{11}a^{9}-\frac{3}{11}a^{8}-\frac{5}{11}a^{7}-\frac{1}{11}a^{6}-\frac{3}{11}a^{5}+\frac{3}{11}a^{4}+\frac{3}{11}a^{3}-\frac{4}{11}a^{2}+\frac{3}{11}a+\frac{1}{11}$, $\frac{1}{11}a^{30}-\frac{2}{11}a^{26}-\frac{5}{11}a^{25}-\frac{2}{11}a^{24}+\frac{2}{11}a^{23}-\frac{4}{11}a^{21}+\frac{4}{11}a^{20}+\frac{1}{11}a^{19}+\frac{5}{11}a^{18}-\frac{3}{11}a^{17}+\frac{4}{11}a^{16}-\frac{5}{11}a^{15}-\frac{2}{11}a^{13}-\frac{1}{11}a^{12}-\frac{2}{11}a^{11}+\frac{2}{11}a^{10}-\frac{3}{11}a^{9}-\frac{5}{11}a^{8}-\frac{1}{11}a^{7}-\frac{3}{11}a^{6}+\frac{3}{11}a^{5}+\frac{3}{11}a^{4}-\frac{4}{11}a^{3}+\frac{3}{11}a^{2}+\frac{1}{11}a$, $\frac{1}{11}a^{31}+\frac{1}{11}a^{26}+\frac{4}{11}a^{25}-\frac{5}{11}a^{24}+\frac{5}{11}a^{23}-\frac{1}{11}a^{22}+\frac{1}{11}a^{21}-\frac{2}{11}a^{19}-\frac{4}{11}a^{18}-\frac{4}{11}a^{17}+\frac{1}{11}a^{16}-\frac{1}{11}a^{15}-\frac{3}{11}a^{14}-\frac{1}{11}a^{13}+\frac{5}{11}a^{12}+\frac{1}{11}a^{11}+\frac{3}{11}a^{10}-\frac{5}{11}a^{9}+\frac{3}{11}a^{8}+\frac{2}{11}a^{7}-\frac{1}{11}a^{6}-\frac{5}{11}a^{5}+\frac{5}{11}a^{4}-\frac{5}{11}a^{3}-\frac{5}{11}a^{2}+\frac{3}{11}a+\frac{4}{11}$, $\frac{1}{11}a^{32}+\frac{1}{11}a^{26}+\frac{3}{11}a^{25}+\frac{3}{11}a^{24}+\frac{2}{11}a^{23}+\frac{5}{11}a^{22}-\frac{4}{11}a^{21}+\frac{4}{11}a^{20}+\frac{5}{11}a^{19}+\frac{2}{11}a^{18}+\frac{5}{11}a^{17}-\frac{4}{11}a^{16}+\frac{3}{11}a^{15}+\frac{5}{11}a^{14}+\frac{5}{11}a^{13}+\frac{3}{11}a^{12}-\frac{2}{11}a^{11}+\frac{3}{11}a^{10}+\frac{3}{11}a^{9}+\frac{2}{11}a^{7}-\frac{3}{11}a^{6}-\frac{2}{11}a^{5}-\frac{4}{11}a^{4}-\frac{1}{11}a^{3}-\frac{5}{11}a^{2}-\frac{3}{11}a-\frac{2}{11}$, $\frac{1}{11}a^{33}-\frac{3}{11}a^{23}+\frac{3}{11}a^{13}-\frac{2}{11}a^{11}-\frac{1}{11}a^{3}+\frac{2}{11}a-\frac{2}{11}$, $\frac{1}{11}a^{34}-\frac{3}{11}a^{24}+\frac{3}{11}a^{14}-\frac{2}{11}a^{12}-\frac{1}{11}a^{4}+\frac{2}{11}a^{2}-\frac{2}{11}a$, $\frac{1}{11}a^{35}-\frac{3}{11}a^{25}+\frac{3}{11}a^{15}-\frac{2}{11}a^{13}-\frac{1}{11}a^{5}+\frac{2}{11}a^{3}-\frac{2}{11}a^{2}$, $\frac{1}{11}a^{36}-\frac{3}{11}a^{26}+\frac{3}{11}a^{16}-\frac{2}{11}a^{14}-\frac{1}{11}a^{6}+\frac{2}{11}a^{4}-\frac{2}{11}a^{3}$, $\frac{1}{11}a^{37}-\frac{2}{11}a^{26}-\frac{2}{11}a^{25}-\frac{5}{11}a^{24}+\frac{2}{11}a^{23}-\frac{1}{11}a^{22}+\frac{1}{11}a^{21}+\frac{4}{11}a^{20}-\frac{5}{11}a^{19}+\frac{4}{11}a^{18}+\frac{2}{11}a^{17}-\frac{2}{11}a^{16}+\frac{2}{11}a^{15}+\frac{4}{11}a^{14}+\frac{5}{11}a^{12}+\frac{4}{11}a^{11}-\frac{2}{11}a^{10}-\frac{5}{11}a^{8}+\frac{1}{11}a^{7}+\frac{5}{11}a^{6}+\frac{1}{11}a^{5}-\frac{5}{11}a^{4}-\frac{1}{11}a^{3}+\frac{2}{11}a^{2}-\frac{1}{11}a-\frac{5}{11}$, $\frac{1}{11}a^{38}+\frac{4}{11}a^{26}+\frac{1}{11}a^{25}-\frac{5}{11}a^{24}+\frac{4}{11}a^{23}+\frac{4}{11}a^{22}+\frac{1}{11}a^{21}+\frac{5}{11}a^{20}-\frac{3}{11}a^{19}+\frac{1}{11}a^{18}+\frac{1}{11}a^{17}-\frac{3}{11}a^{16}+\frac{3}{11}a^{15}-\frac{1}{11}a^{14}+\frac{5}{11}a^{13}-\frac{3}{11}a^{11}-\frac{5}{11}a^{10}-\frac{5}{11}a^{9}+\frac{5}{11}a^{8}-\frac{1}{11}a^{7}-\frac{3}{11}a^{6}-\frac{2}{11}a^{5}-\frac{3}{11}a^{4}+\frac{5}{11}a^{3}+\frac{4}{11}a^{2}-\frac{2}{11}a+\frac{4}{11}$, $\frac{1}{11}a^{39}+\frac{5}{11}a^{25}-\frac{4}{11}a^{24}+\frac{5}{11}a^{23}-\frac{5}{11}a^{22}-\frac{1}{11}a^{20}+\frac{4}{11}a^{19}+\frac{3}{11}a^{18}+\frac{2}{11}a^{17}+\frac{2}{11}a^{16}+\frac{1}{11}a^{15}-\frac{4}{11}a^{14}+\frac{5}{11}a^{12}-\frac{3}{11}a^{11}+\frac{5}{11}a^{10}+\frac{5}{11}a^{9}+\frac{2}{11}a^{8}-\frac{2}{11}a^{7}-\frac{5}{11}a^{6}+\frac{2}{11}a^{5}-\frac{2}{11}a^{4}-\frac{2}{11}a^{3}-\frac{1}{11}a^{2}-\frac{2}{11}a+\frac{3}{11}$, $\frac{1}{11}a^{40}+\frac{5}{11}a^{26}-\frac{4}{11}a^{25}+\frac{5}{11}a^{24}-\frac{5}{11}a^{23}-\frac{1}{11}a^{21}+\frac{4}{11}a^{20}+\frac{3}{11}a^{19}+\frac{2}{11}a^{18}+\frac{2}{11}a^{17}+\frac{1}{11}a^{16}-\frac{4}{11}a^{15}+\frac{5}{11}a^{13}-\frac{3}{11}a^{12}+\frac{5}{11}a^{11}+\frac{5}{11}a^{10}+\frac{2}{11}a^{9}-\frac{2}{11}a^{8}-\frac{5}{11}a^{7}+\frac{2}{11}a^{6}-\frac{2}{11}a^{5}-\frac{2}{11}a^{4}-\frac{1}{11}a^{3}-\frac{2}{11}a^{2}+\frac{3}{11}a$, $\frac{1}{6589}a^{41}-\frac{82}{6589}a^{40}+\frac{299}{6589}a^{39}+\frac{217}{6589}a^{38}-\frac{276}{6589}a^{37}-\frac{2}{599}a^{36}+\frac{20}{599}a^{35}-\frac{150}{6589}a^{34}-\frac{83}{6589}a^{33}-\frac{8}{6589}a^{32}+\frac{223}{6589}a^{31}-\frac{106}{6589}a^{30}+\frac{233}{6589}a^{29}+\frac{179}{6589}a^{28}+\frac{183}{6589}a^{27}-\frac{3111}{6589}a^{26}-\frac{1975}{6589}a^{25}-\frac{288}{599}a^{24}+\frac{1085}{6589}a^{23}-\frac{1737}{6589}a^{22}-\frac{143}{599}a^{21}+\frac{1009}{6589}a^{20}+\frac{812}{6589}a^{19}+\frac{2028}{6589}a^{18}-\frac{786}{6589}a^{17}-\frac{2232}{6589}a^{16}-\frac{2853}{6589}a^{15}-\frac{757}{6589}a^{14}-\frac{2340}{6589}a^{13}-\frac{59}{6589}a^{12}-\frac{1258}{6589}a^{11}-\frac{2998}{6589}a^{10}+\frac{3283}{6589}a^{9}-\frac{2395}{6589}a^{8}-\frac{2987}{6589}a^{7}+\frac{1971}{6589}a^{6}-\frac{1051}{6589}a^{5}+\frac{2572}{6589}a^{4}+\frac{2996}{6589}a^{3}-\frac{2069}{6589}a^{2}-\frac{549}{6589}a-\frac{1370}{6589}$, $\frac{1}{16887607}a^{42}+\frac{1176}{16887607}a^{41}+\frac{363764}{16887607}a^{40}+\frac{508738}{16887607}a^{39}+\frac{1173}{72479}a^{38}-\frac{56116}{16887607}a^{37}+\frac{161828}{16887607}a^{36}-\frac{16348}{1535237}a^{35}-\frac{45424}{1535237}a^{34}+\frac{368189}{16887607}a^{33}+\frac{398677}{16887607}a^{32}-\frac{90952}{16887607}a^{31}-\frac{472149}{16887607}a^{30}+\frac{737751}{16887607}a^{29}+\frac{331388}{16887607}a^{28}+\frac{1879}{16887607}a^{27}-\frac{439008}{1535237}a^{26}-\frac{540968}{16887607}a^{25}-\frac{7692668}{16887607}a^{24}+\frac{733273}{1535237}a^{23}-\frac{3252939}{16887607}a^{22}-\frac{4214491}{16887607}a^{21}-\frac{985700}{16887607}a^{20}+\frac{723359}{1535237}a^{19}-\frac{2287685}{16887607}a^{18}+\frac{5021143}{16887607}a^{17}-\frac{2684919}{16887607}a^{16}+\frac{4068384}{16887607}a^{15}+\frac{5527732}{16887607}a^{14}+\frac{353117}{16887607}a^{13}+\frac{597148}{1535237}a^{12}+\frac{6597377}{16887607}a^{11}-\frac{1524946}{16887607}a^{10}-\frac{5552221}{16887607}a^{9}-\frac{5323844}{16887607}a^{8}+\frac{6223066}{16887607}a^{7}+\frac{4395267}{16887607}a^{6}-\frac{1071600}{16887607}a^{5}+\frac{7303386}{16887607}a^{4}+\frac{5823266}{16887607}a^{3}-\frac{4238621}{16887607}a^{2}+\frac{6941644}{16887607}a-\frac{3500094}{16887607}$, $\frac{1}{55423825828261}a^{43}-\frac{1144799}{55423825828261}a^{42}+\frac{21233722}{5038529620751}a^{41}+\frac{1756808198246}{55423825828261}a^{40}+\frac{425335439340}{55423825828261}a^{39}-\frac{401703172080}{55423825828261}a^{38}-\frac{1374557004014}{55423825828261}a^{37}+\frac{2283058121291}{55423825828261}a^{36}+\frac{10207521068}{458048147341}a^{35}+\frac{1594776002168}{55423825828261}a^{34}+\frac{6168155580}{5038529620751}a^{33}-\frac{1921990259107}{55423825828261}a^{32}+\frac{2416850839266}{55423825828261}a^{31}+\frac{1378357741168}{55423825828261}a^{30}+\frac{2380143117270}{55423825828261}a^{29}+\frac{1441983625178}{55423825828261}a^{28}-\frac{1090779479349}{55423825828261}a^{27}+\frac{14878793572765}{55423825828261}a^{26}-\frac{1606400349997}{5038529620751}a^{25}-\frac{26478354856968}{55423825828261}a^{24}+\frac{14635658190860}{55423825828261}a^{23}-\frac{23354690752791}{55423825828261}a^{22}+\frac{13323770062968}{55423825828261}a^{21}-\frac{24387035578229}{55423825828261}a^{20}-\frac{5611711289507}{55423825828261}a^{19}-\frac{16301288426701}{55423825828261}a^{18}+\frac{8598561458126}{55423825828261}a^{17}+\frac{25060697379733}{55423825828261}a^{16}-\frac{242256512800}{55423825828261}a^{15}-\frac{1208063827849}{5038529620751}a^{14}-\frac{10061753890873}{55423825828261}a^{13}+\frac{1061004525279}{55423825828261}a^{12}-\frac{13070869957626}{55423825828261}a^{11}-\frac{20290443681735}{55423825828261}a^{10}+\frac{14932553963386}{55423825828261}a^{9}+\frac{25541975798583}{55423825828261}a^{8}+\frac{9259843119693}{55423825828261}a^{7}-\frac{22488848232835}{55423825828261}a^{6}+\frac{1705292860252}{55423825828261}a^{5}-\frac{12021269908057}{55423825828261}a^{4}-\frac{15088605315554}{55423825828261}a^{3}-\frac{9036857420122}{55423825828261}a^{2}+\frac{14284039851283}{55423825828261}a+\frac{22829998495251}{55423825828261}$, $\frac{1}{61\!\cdots\!47}a^{44}-\frac{19\!\cdots\!28}{61\!\cdots\!47}a^{43}+\frac{16\!\cdots\!04}{61\!\cdots\!47}a^{42}+\frac{30\!\cdots\!76}{61\!\cdots\!47}a^{41}+\frac{52\!\cdots\!41}{56\!\cdots\!77}a^{40}+\frac{15\!\cdots\!78}{61\!\cdots\!47}a^{39}+\frac{13\!\cdots\!94}{61\!\cdots\!47}a^{38}+\frac{19\!\cdots\!96}{61\!\cdots\!47}a^{37}-\frac{27\!\cdots\!04}{61\!\cdots\!47}a^{36}-\frac{17\!\cdots\!85}{61\!\cdots\!47}a^{35}+\frac{24\!\cdots\!41}{61\!\cdots\!47}a^{34}-\frac{94\!\cdots\!51}{61\!\cdots\!47}a^{33}-\frac{19\!\cdots\!68}{61\!\cdots\!47}a^{32}-\frac{23\!\cdots\!40}{61\!\cdots\!47}a^{31}+\frac{25\!\cdots\!08}{61\!\cdots\!47}a^{30}-\frac{15\!\cdots\!85}{61\!\cdots\!47}a^{29}-\frac{21\!\cdots\!29}{61\!\cdots\!47}a^{28}+\frac{23\!\cdots\!35}{61\!\cdots\!47}a^{27}+\frac{26\!\cdots\!88}{61\!\cdots\!47}a^{26}-\frac{30\!\cdots\!70}{61\!\cdots\!47}a^{25}-\frac{30\!\cdots\!23}{61\!\cdots\!47}a^{24}+\frac{14\!\cdots\!28}{61\!\cdots\!47}a^{23}+\frac{42\!\cdots\!32}{61\!\cdots\!47}a^{22}+\frac{58\!\cdots\!93}{61\!\cdots\!47}a^{21}+\frac{29\!\cdots\!98}{61\!\cdots\!47}a^{20}+\frac{10\!\cdots\!97}{61\!\cdots\!47}a^{19}-\frac{10\!\cdots\!52}{61\!\cdots\!47}a^{18}+\frac{15\!\cdots\!67}{61\!\cdots\!47}a^{17}+\frac{24\!\cdots\!54}{61\!\cdots\!47}a^{16}+\frac{23\!\cdots\!69}{61\!\cdots\!47}a^{15}-\frac{28\!\cdots\!12}{61\!\cdots\!47}a^{14}-\frac{15\!\cdots\!19}{56\!\cdots\!77}a^{13}+\frac{30\!\cdots\!70}{61\!\cdots\!47}a^{12}+\frac{70\!\cdots\!15}{56\!\cdots\!77}a^{11}+\frac{17\!\cdots\!14}{61\!\cdots\!47}a^{10}-\frac{14\!\cdots\!52}{61\!\cdots\!47}a^{9}+\frac{24\!\cdots\!60}{56\!\cdots\!77}a^{8}-\frac{46\!\cdots\!24}{61\!\cdots\!47}a^{7}-\frac{11\!\cdots\!68}{61\!\cdots\!47}a^{6}+\frac{23\!\cdots\!03}{61\!\cdots\!47}a^{5}-\frac{26\!\cdots\!80}{56\!\cdots\!77}a^{4}-\frac{82\!\cdots\!88}{61\!\cdots\!47}a^{3}-\frac{23\!\cdots\!04}{61\!\cdots\!47}a^{2}-\frac{34\!\cdots\!07}{61\!\cdots\!47}a-\frac{88\!\cdots\!64}{10\!\cdots\!53}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $44$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
Galois group
$C_3\times C_{15}$ (as 45T2):
| An abelian group of order 45 |
| The 45 conjugacy class representatives for $C_3\times C_{15}$ |
| Character table for $C_3\times C_{15}$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $15^{3}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{15}$ | R | $15^{3}$ | $15^{3}$ | $15^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{15}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{15}$ | ${\href{/padicField/53.5.0.1}{5} }^{9}$ | $15^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.9.6.1 | $x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 13.9.6.1 | $x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 13.9.6.1 | $x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 13.9.6.1 | $x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 13.9.6.1 | $x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
|
\(61\)
| Deg $45$ | $15$ | $3$ | $42$ |