Normalized defining polynomial
\( x^{17} - 7 x^{16} + 22 x^{15} - 34 x^{14} + 16 x^{13} + 8 x^{12} - 48 x^{11} + 12 x^{10} - 76 x^{8} + \cdots - 3 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1863207356329472909377536\) \(\medspace = 2^{41}\cdot 3^{25}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(26.77\) | 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))
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Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{6}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{4}{9}a^{8}+\frac{4}{9}a^{7}+\frac{1}{9}a^{6}-\frac{2}{9}a^{5}-\frac{1}{3}a^{4}+\frac{1}{9}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{4}{9}a^{7}+\frac{4}{9}a^{6}-\frac{2}{9}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{12}-\frac{1}{9}a^{6}-\frac{1}{3}a^{4}-\frac{4}{9}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{8488134}a^{16}-\frac{106649}{4244067}a^{15}+\frac{1764}{471563}a^{14}+\frac{85970}{4244067}a^{13}-\frac{371570}{4244067}a^{12}-\frac{232907}{1414689}a^{11}+\frac{162694}{1414689}a^{10}+\frac{12476}{83217}a^{9}+\frac{3831}{27739}a^{8}-\frac{1545032}{4244067}a^{7}-\frac{1894081}{4244067}a^{6}-\frac{203957}{471563}a^{5}+\frac{779113}{4244067}a^{4}+\frac{1010333}{4244067}a^{3}-\frac{496954}{1414689}a^{2}+\frac{80763}{471563}a+\frac{109019}{2829378}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{71965}{1414689}a^{16}+\frac{1373197}{4244067}a^{15}-\frac{1253071}{1414689}a^{14}+\frac{3754001}{4244067}a^{13}+\frac{4103314}{4244067}a^{12}-\frac{10925566}{4244067}a^{11}+\frac{16156070}{4244067}a^{10}+\frac{398755}{249651}a^{9}-\frac{471239}{249651}a^{8}+\frac{34655582}{4244067}a^{7}+\frac{3149704}{4244067}a^{6}+\frac{3320315}{4244067}a^{5}+\frac{13151074}{1414689}a^{4}-\frac{5925392}{4244067}a^{3}+\frac{2097955}{1414689}a^{2}+\frac{4414559}{1414689}a-\frac{1419238}{1414689}$, $\frac{65611}{1414689}a^{16}-\frac{831344}{4244067}a^{15}+\frac{688540}{4244067}a^{14}+\frac{3633577}{4244067}a^{13}-\frac{9308347}{4244067}a^{12}-\frac{3286}{4244067}a^{11}-\frac{1299805}{4244067}a^{10}-\frac{908969}{249651}a^{9}+\frac{45751}{249651}a^{8}-\frac{1780534}{4244067}a^{7}-\frac{14022236}{4244067}a^{6}+\frac{19803958}{4244067}a^{5}+\frac{6666184}{4244067}a^{4}+\frac{269618}{471563}a^{3}+\frac{4391572}{1414689}a^{2}+\frac{251279}{1414689}a-\frac{293538}{471563}$, $\frac{233384}{4244067}a^{16}-\frac{2150152}{4244067}a^{15}+\frac{8284357}{4244067}a^{14}-\frac{16584793}{4244067}a^{13}+\frac{4753342}{1414689}a^{12}+\frac{1326769}{4244067}a^{11}-\frac{15076037}{4244067}a^{10}+\frac{1668410}{249651}a^{9}+\frac{663371}{249651}a^{8}-\frac{297853}{4244067}a^{7}+\frac{63004457}{4244067}a^{6}+\frac{4572089}{1414689}a^{5}+\frac{28640594}{4244067}a^{4}+\frac{48862516}{4244067}a^{3}+\frac{3387532}{1414689}a^{2}+\frac{1780890}{471563}a+\frac{3881135}{1414689}$, $\frac{8268277}{4244067}a^{16}-\frac{18232340}{1414689}a^{15}+\frac{160922171}{4244067}a^{14}-\frac{73037681}{1414689}a^{13}+\frac{45953729}{4244067}a^{12}+\frac{90976063}{4244067}a^{11}-\frac{370799195}{4244067}a^{10}-\frac{2813053}{249651}a^{9}-\frac{620119}{249651}a^{8}-\frac{217432249}{1414689}a^{7}+\frac{117058630}{4244067}a^{6}-\frac{230538256}{4244067}a^{5}-\frac{384470531}{4244067}a^{4}+\frac{109925465}{4244067}a^{3}-\frac{12871860}{471563}a^{2}-\frac{6757260}{471563}a+\frac{16518586}{1414689}$, $\frac{2171021}{4244067}a^{16}+\frac{5044279}{1414689}a^{15}-\frac{46449500}{4244067}a^{14}+\frac{66637939}{4244067}a^{13}-\frac{16125992}{4244067}a^{12}-\frac{38278901}{4244067}a^{11}+\frac{100869673}{4244067}a^{10}-\frac{495019}{249651}a^{9}-\frac{1744423}{249651}a^{8}+\frac{18176687}{471563}a^{7}-\frac{79880873}{4244067}a^{6}+\frac{3034328}{471563}a^{5}+\frac{31501829}{1414689}a^{4}-\frac{6568559}{471563}a^{3}+\frac{2420615}{471563}a^{2}+\frac{1958467}{471563}a-\frac{2789880}{471563}$, $\frac{1221494}{4244067}a^{16}+\frac{774517}{471563}a^{15}-\frac{15797006}{4244067}a^{14}+\frac{6709336}{4244067}a^{13}+\frac{34619653}{4244067}a^{12}-\frac{36811907}{4244067}a^{11}+\frac{49685320}{4244067}a^{10}+\frac{3405950}{249651}a^{9}-\frac{1144714}{249651}a^{8}+\frac{26620780}{1414689}a^{7}+\frac{49543450}{4244067}a^{6}-\frac{14043152}{1414689}a^{5}+\frac{24329734}{1414689}a^{4}-\frac{1615223}{471563}a^{3}-\frac{3444887}{471563}a^{2}+\frac{1748734}{471563}a-\frac{145230}{471563}$, $\frac{6419725}{4244067}a^{16}-\frac{45036521}{4244067}a^{15}+\frac{141340031}{4244067}a^{14}-\frac{217665352}{4244067}a^{13}+\frac{11564326}{471563}a^{12}+\frac{10495216}{1414689}a^{11}-\frac{27981178}{471563}a^{10}+\frac{741334}{83217}a^{9}+\frac{141248}{83217}a^{8}-\frac{449070811}{4244067}a^{7}+\frac{246828731}{4244067}a^{6}-\frac{236262023}{4244067}a^{5}-\frac{175880210}{4244067}a^{4}+\frac{25321577}{1414689}a^{3}-\frac{36193385}{1414689}a^{2}-\frac{6356473}{1414689}a+\frac{2634548}{471563}$, $\frac{44581}{249651}a^{16}+\frac{108938}{249651}a^{15}+\frac{345865}{249651}a^{14}-\frac{2330651}{249651}a^{13}+\frac{1512527}{83217}a^{12}-\frac{236341}{27739}a^{11}+\frac{847531}{83217}a^{10}+\frac{1847818}{83217}a^{9}+\frac{167381}{27739}a^{8}+\frac{4133281}{249651}a^{7}+\frac{9345406}{249651}a^{6}-\frac{3298429}{249651}a^{5}+\frac{6217232}{249651}a^{4}-\frac{154874}{83217}a^{3}-\frac{311272}{83217}a^{2}+\frac{272338}{83217}a-\frac{54848}{27739}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 1860535.6396997238 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 1860535.6396997238 \cdot 1}{2\cdot\sqrt{1863207356329472909377536}}\cr\approx \mathstrut & 3.31090214631167 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 355687428096000 |
The 297 conjugacy class representatives for $S_{17}$ |
Character table for $S_{17}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | $17$ | $16{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.7.0.1}{7} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $17$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $16$ | $16$ | $1$ | $41$ | ||||
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
3.12.21.8 | $x^{12} + 3 x^{10} + 6 x^{9} + 3 x^{6} + 15 x^{3} + 24$ | $12$ | $1$ | $21$ | 12T118 | $[2, 9/4, 9/4]_{4}^{2}$ |