Normalized defining polynomial
\( x^{17} - 14 x^{15} - x^{14} + 76 x^{13} + 11 x^{12} - 207 x^{11} - 49 x^{10} + 312 x^{9} + 104 x^{8} + \cdots - 1 \)
Invariants
Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[9, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(7367931847600131843027939152\) \(\medspace = 2^{4}\cdot 460495740475008240189246197\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(43.58\) | 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: | $2^{4/5}460495740475008240189246197^{1/2}\approx 37362575463836.61$ | ||
Ramified primes: | \(2\), \(460495740475008240189246197\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{46049\!\cdots\!46197}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $12$ | 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: | $a$, $a^{3}-3a+1$, $10a^{16}+3a^{15}-139a^{14}-52a^{13}+743a^{12}+336a^{11}-1961a^{10}-1089a^{9}+2768a^{8}+1884a^{7}-2202a^{6}-1653a^{5}+1026a^{4}+701a^{3}-272a^{2}-107a+35$, $13a^{16}+7a^{15}-180a^{14}-109a^{13}+953a^{12}+646a^{11}-2464a^{10}-1925a^{9}+3321a^{8}+3088a^{7}-2399a^{6}-2576a^{5}+974a^{4}+1052a^{3}-229a^{2}-154a+31$, $a^{16}-14a^{14}-a^{13}+76a^{12}+11a^{11}-207a^{10}-49a^{9}+312a^{8}+104a^{7}-281a^{6}-99a^{5}+156a^{4}+40a^{3}-50a^{2}-4a+8$, $a^{16}-a^{15}-13a^{14}+12a^{13}+64a^{12}-53a^{11}-154a^{10}+105a^{9}+207a^{8}-103a^{7}-178a^{6}+79a^{5}+77a^{4}-37a^{3}-13a^{2}+10a-2$, $14a^{16}+7a^{15}-194a^{14}-110a^{13}+1029a^{12}+657a^{11}-2671a^{10}-1974a^{9}+3633a^{8}+3192a^{7}-2680a^{6}-2675a^{5}+1130a^{4}+1093a^{3}-279a^{2}-160a+39$, $4a^{16}+3a^{15}-55a^{14}-45a^{13}+287a^{12}+258a^{11}-721a^{10}-741a^{9}+914a^{8}+1141a^{7}-575a^{6}-916a^{5}+175a^{4}+365a^{3}-29a^{2}-52a+6$, $15a^{16}+7a^{15}-207a^{14}-112a^{13}+1092a^{12}+679a^{11}-2813a^{10}-2066a^{9}+3783a^{8}+3366a^{7}-2736a^{6}-2819a^{5}+1100a^{4}+1153a^{3}-249a^{2}-173a+35$, $8a^{16}+4a^{15}-110a^{14}-63a^{13}+576a^{12}+376a^{11}-1462a^{10}-1123a^{9}+1909a^{8}+1787a^{7}-1307a^{6}-1447a^{5}+481a^{4}+563a^{3}-89a^{2}-78a+10$, $10a^{16}+8a^{15}-140a^{14}-118a^{13}+750a^{12}+665a^{11}-1960a^{10}-1879a^{9}+2643a^{8}+2879a^{7}-1858a^{6}-2362a^{5}+745a^{4}+950a^{3}-181a^{2}-136a+27$, $15a^{16}+7a^{15}-207a^{14}-112a^{13}+1092a^{12}+679a^{11}-2813a^{10}-2067a^{9}+3785a^{8}+3372a^{7}-2747a^{6}-2830a^{5}+1117a^{4}+1162a^{3}-261a^{2}-174a+37$ (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: | \( 93831886.8506 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{4}\cdot 93831886.8506 \cdot 1}{2\cdot\sqrt{7367931847600131843027939152}}\cr\approx \mathstrut & 0.436151216663 \end{aligned}\] (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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
2.5.4.1 | $x^{5} + 2$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
2.7.0.1 | $x^{7} + x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(460\!\cdots\!197\) | $\Q_{46\!\cdots\!97}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{46\!\cdots\!97}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{46\!\cdots\!97}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |