Normalized defining polynomial
\( x^{17} - 34 x^{14} - 68 x^{13} + 17 x^{12} + 323 x^{11} + 884 x^{10} + 1241 x^{9} + 1394 x^{8} + \cdots + 57 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(92267897090016343666010049\) \(\medspace = 3^{8}\cdot 17^{18}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.68\) | 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: | $3^{1/2}17^{151/136}\approx 40.245824295791174$ | ||
Ramified primes: | \(3\), \(17\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{26}a^{15}-\frac{1}{26}a^{14}-\frac{1}{26}a^{13}-\frac{5}{26}a^{12}+\frac{2}{13}a^{11}+\frac{5}{13}a^{10}+\frac{3}{13}a^{9}-\frac{1}{2}a^{8}-\frac{3}{13}a^{7}-\frac{2}{13}a^{6}+\frac{5}{26}a^{5}-\frac{11}{26}a^{4}-\frac{6}{13}a^{3}-\frac{1}{26}a^{2}-\frac{4}{13}a-\frac{2}{13}$, $\frac{1}{29\!\cdots\!78}a^{16}+\frac{40\!\cdots\!43}{22\!\cdots\!06}a^{15}+\frac{22\!\cdots\!59}{14\!\cdots\!89}a^{14}-\frac{11\!\cdots\!85}{14\!\cdots\!89}a^{13}+\frac{47\!\cdots\!65}{29\!\cdots\!78}a^{12}+\frac{21\!\cdots\!35}{14\!\cdots\!89}a^{11}-\frac{65\!\cdots\!70}{14\!\cdots\!89}a^{10}+\frac{10\!\cdots\!11}{29\!\cdots\!78}a^{9}+\frac{10\!\cdots\!83}{29\!\cdots\!78}a^{8}+\frac{26\!\cdots\!67}{14\!\cdots\!89}a^{7}-\frac{56\!\cdots\!59}{29\!\cdots\!78}a^{6}-\frac{65\!\cdots\!63}{14\!\cdots\!89}a^{5}-\frac{78\!\cdots\!11}{29\!\cdots\!78}a^{4}+\frac{97\!\cdots\!75}{22\!\cdots\!06}a^{3}-\frac{98\!\cdots\!35}{29\!\cdots\!78}a^{2}+\frac{13\!\cdots\!91}{14\!\cdots\!89}a+\frac{11\!\cdots\!45}{29\!\cdots\!78}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{43\!\cdots\!63}{29\!\cdots\!78}a^{16}-\frac{32\!\cdots\!57}{14\!\cdots\!89}a^{15}-\frac{61\!\cdots\!01}{29\!\cdots\!78}a^{14}-\frac{14\!\cdots\!13}{29\!\cdots\!78}a^{13}-\frac{82\!\cdots\!69}{29\!\cdots\!78}a^{12}+\frac{51\!\cdots\!81}{29\!\cdots\!78}a^{11}+\frac{53\!\cdots\!15}{14\!\cdots\!89}a^{10}+\frac{74\!\cdots\!49}{14\!\cdots\!89}a^{9}+\frac{19\!\cdots\!42}{14\!\cdots\!89}a^{8}+\frac{16\!\cdots\!03}{29\!\cdots\!78}a^{7}-\frac{62\!\cdots\!04}{14\!\cdots\!89}a^{6}-\frac{11\!\cdots\!25}{14\!\cdots\!89}a^{5}-\frac{84\!\cdots\!79}{29\!\cdots\!78}a^{4}+\frac{86\!\cdots\!27}{29\!\cdots\!78}a^{3}-\frac{17\!\cdots\!19}{14\!\cdots\!89}a^{2}+\frac{40\!\cdots\!90}{14\!\cdots\!89}a-\frac{21\!\cdots\!44}{14\!\cdots\!89}$, $\frac{11\!\cdots\!43}{14\!\cdots\!89}a^{16}-\frac{20\!\cdots\!08}{14\!\cdots\!89}a^{15}-\frac{12\!\cdots\!82}{14\!\cdots\!89}a^{14}-\frac{37\!\cdots\!26}{14\!\cdots\!89}a^{13}-\frac{13\!\cdots\!95}{29\!\cdots\!78}a^{12}+\frac{14\!\cdots\!75}{29\!\cdots\!78}a^{11}+\frac{41\!\cdots\!34}{14\!\cdots\!89}a^{10}+\frac{12\!\cdots\!75}{22\!\cdots\!06}a^{9}+\frac{81\!\cdots\!57}{14\!\cdots\!89}a^{8}+\frac{13\!\cdots\!25}{29\!\cdots\!78}a^{7}+\frac{73\!\cdots\!05}{29\!\cdots\!78}a^{6}+\frac{98\!\cdots\!67}{29\!\cdots\!78}a^{5}+\frac{86\!\cdots\!11}{29\!\cdots\!78}a^{4}+\frac{48\!\cdots\!54}{14\!\cdots\!89}a^{3}+\frac{24\!\cdots\!33}{14\!\cdots\!89}a^{2}+\frac{88\!\cdots\!06}{14\!\cdots\!89}a+\frac{40\!\cdots\!77}{22\!\cdots\!06}$, $\frac{12\!\cdots\!95}{29\!\cdots\!78}a^{16}-\frac{66\!\cdots\!64}{14\!\cdots\!89}a^{15}-\frac{21\!\cdots\!38}{14\!\cdots\!89}a^{14}-\frac{15\!\cdots\!52}{11\!\cdots\!53}a^{13}-\frac{19\!\cdots\!68}{14\!\cdots\!89}a^{12}+\frac{62\!\cdots\!84}{14\!\cdots\!89}a^{11}+\frac{16\!\cdots\!18}{14\!\cdots\!89}a^{10}+\frac{26\!\cdots\!33}{14\!\cdots\!89}a^{9}+\frac{18\!\cdots\!67}{14\!\cdots\!89}a^{8}+\frac{14\!\cdots\!96}{11\!\cdots\!53}a^{7}+\frac{83\!\cdots\!53}{14\!\cdots\!89}a^{6}+\frac{22\!\cdots\!15}{14\!\cdots\!89}a^{5}+\frac{71\!\cdots\!51}{14\!\cdots\!89}a^{4}+\frac{16\!\cdots\!96}{14\!\cdots\!89}a^{3}+\frac{17\!\cdots\!86}{14\!\cdots\!89}a^{2}+\frac{88\!\cdots\!33}{29\!\cdots\!78}a+\frac{55\!\cdots\!12}{14\!\cdots\!89}$, $\frac{50\!\cdots\!87}{29\!\cdots\!78}a^{16}+\frac{15\!\cdots\!65}{14\!\cdots\!89}a^{15}-\frac{54\!\cdots\!91}{29\!\cdots\!78}a^{14}-\frac{66\!\cdots\!89}{11\!\cdots\!53}a^{13}-\frac{22\!\cdots\!62}{14\!\cdots\!89}a^{12}+\frac{26\!\cdots\!37}{14\!\cdots\!89}a^{11}+\frac{20\!\cdots\!73}{29\!\cdots\!78}a^{10}+\frac{26\!\cdots\!52}{14\!\cdots\!89}a^{9}+\frac{35\!\cdots\!98}{14\!\cdots\!89}a^{8}+\frac{46\!\cdots\!79}{22\!\cdots\!06}a^{7}+\frac{34\!\cdots\!53}{29\!\cdots\!78}a^{6}+\frac{12\!\cdots\!15}{14\!\cdots\!89}a^{5}+\frac{33\!\cdots\!69}{29\!\cdots\!78}a^{4}+\frac{18\!\cdots\!36}{14\!\cdots\!89}a^{3}+\frac{26\!\cdots\!41}{29\!\cdots\!78}a^{2}+\frac{38\!\cdots\!50}{14\!\cdots\!89}a+\frac{11\!\cdots\!87}{29\!\cdots\!78}$, $\frac{21\!\cdots\!85}{29\!\cdots\!78}a^{16}+\frac{26\!\cdots\!59}{29\!\cdots\!78}a^{15}-\frac{36\!\cdots\!01}{29\!\cdots\!78}a^{14}-\frac{70\!\cdots\!83}{29\!\cdots\!78}a^{13}-\frac{89\!\cdots\!15}{11\!\cdots\!53}a^{12}-\frac{12\!\cdots\!04}{14\!\cdots\!89}a^{11}+\frac{46\!\cdots\!02}{14\!\cdots\!89}a^{10}+\frac{24\!\cdots\!27}{29\!\cdots\!78}a^{9}+\frac{18\!\cdots\!09}{14\!\cdots\!89}a^{8}+\frac{18\!\cdots\!94}{14\!\cdots\!89}a^{7}+\frac{30\!\cdots\!67}{29\!\cdots\!78}a^{6}+\frac{21\!\cdots\!67}{29\!\cdots\!78}a^{5}+\frac{92\!\cdots\!03}{11\!\cdots\!53}a^{4}+\frac{18\!\cdots\!95}{29\!\cdots\!78}a^{3}+\frac{10\!\cdots\!28}{14\!\cdots\!89}a^{2}+\frac{33\!\cdots\!65}{11\!\cdots\!53}a+\frac{31\!\cdots\!03}{14\!\cdots\!89}$, $\frac{32\!\cdots\!89}{29\!\cdots\!78}a^{16}-\frac{85\!\cdots\!13}{29\!\cdots\!78}a^{15}-\frac{13\!\cdots\!34}{11\!\cdots\!53}a^{14}-\frac{53\!\cdots\!03}{14\!\cdots\!89}a^{13}-\frac{96\!\cdots\!22}{14\!\cdots\!89}a^{12}+\frac{11\!\cdots\!11}{14\!\cdots\!89}a^{11}+\frac{56\!\cdots\!45}{14\!\cdots\!89}a^{10}+\frac{11\!\cdots\!35}{14\!\cdots\!89}a^{9}+\frac{12\!\cdots\!55}{14\!\cdots\!89}a^{8}+\frac{10\!\cdots\!38}{14\!\cdots\!89}a^{7}+\frac{61\!\cdots\!82}{14\!\cdots\!89}a^{6}+\frac{68\!\cdots\!03}{14\!\cdots\!89}a^{5}+\frac{66\!\cdots\!66}{14\!\cdots\!89}a^{4}+\frac{67\!\cdots\!15}{14\!\cdots\!89}a^{3}+\frac{44\!\cdots\!96}{14\!\cdots\!89}a^{2}+\frac{31\!\cdots\!35}{29\!\cdots\!78}a+\frac{97\!\cdots\!13}{29\!\cdots\!78}$, $\frac{34\!\cdots\!29}{22\!\cdots\!06}a^{16}-\frac{52\!\cdots\!57}{14\!\cdots\!89}a^{15}+\frac{11\!\cdots\!29}{29\!\cdots\!78}a^{14}-\frac{14\!\cdots\!17}{29\!\cdots\!78}a^{13}+\frac{20\!\cdots\!74}{14\!\cdots\!89}a^{12}+\frac{73\!\cdots\!07}{29\!\cdots\!78}a^{11}+\frac{41\!\cdots\!10}{14\!\cdots\!89}a^{10}+\frac{14\!\cdots\!41}{29\!\cdots\!78}a^{9}-\frac{17\!\cdots\!73}{22\!\cdots\!06}a^{8}-\frac{21\!\cdots\!87}{29\!\cdots\!78}a^{7}-\frac{25\!\cdots\!01}{29\!\cdots\!78}a^{6}+\frac{10\!\cdots\!36}{14\!\cdots\!89}a^{5}-\frac{82\!\cdots\!43}{14\!\cdots\!89}a^{4}-\frac{13\!\cdots\!19}{14\!\cdots\!89}a^{3}-\frac{21\!\cdots\!61}{29\!\cdots\!78}a^{2}-\frac{45\!\cdots\!03}{29\!\cdots\!78}a-\frac{18\!\cdots\!66}{14\!\cdots\!89}$, $\frac{14\!\cdots\!43}{29\!\cdots\!78}a^{16}-\frac{28\!\cdots\!66}{14\!\cdots\!89}a^{15}+\frac{65\!\cdots\!17}{22\!\cdots\!06}a^{14}-\frac{52\!\cdots\!07}{29\!\cdots\!78}a^{13}+\frac{82\!\cdots\!55}{29\!\cdots\!78}a^{12}+\frac{63\!\cdots\!66}{14\!\cdots\!89}a^{11}-\frac{27\!\cdots\!94}{14\!\cdots\!89}a^{10}+\frac{12\!\cdots\!12}{14\!\cdots\!89}a^{9}-\frac{12\!\cdots\!93}{29\!\cdots\!78}a^{8}-\frac{18\!\cdots\!85}{14\!\cdots\!89}a^{7}-\frac{14\!\cdots\!11}{14\!\cdots\!89}a^{6}-\frac{30\!\cdots\!87}{29\!\cdots\!78}a^{5}-\frac{52\!\cdots\!03}{29\!\cdots\!78}a^{4}-\frac{65\!\cdots\!47}{14\!\cdots\!89}a^{3}-\frac{26\!\cdots\!03}{29\!\cdots\!78}a^{2}+\frac{43\!\cdots\!29}{29\!\cdots\!78}a-\frac{94\!\cdots\!19}{29\!\cdots\!78}$ | 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: | \( 4992205.48619 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 4992205.48619 \cdot 1}{2\cdot\sqrt{92267897090016343666010049}}\cr\approx \mathstrut & 1.26242646052 \end{aligned}\]
Galois group
$C_{17}:C_8$ (as 17T4):
A solvable group of order 136 |
The 10 conjugacy class representatives for $C_{17}:C_{8}$ |
Character table for $C_{17}:C_{8}$ |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $17$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.8.4.2 | $x^{8} - 6 x^{6} + 162$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
3.8.4.2 | $x^{8} - 6 x^{6} + 162$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
\(17\) | 17.17.18.4 | $x^{17} + 136 x^{2} + 17$ | $17$ | $1$ | $18$ | $C_{17}:C_{8}$ | $[9/8]_{8}$ |