Normalized defining polynomial
\( x^{17} + 17x^{15} + 119x^{13} + 442x^{11} + 935x^{9} + 1122x^{7} + 714x^{5} + 204x^{3} + 17x - 1 \)
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: | \(323140727299350298506640625\) \(\medspace = 5^{8}\cdot 17^{17}\) | 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: | \(36.26\) | 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: | $5^{1/2}17^{287/272}\approx 44.44158735609407$ | ||
Ramified primes: | \(5\), \(17\) | 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: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $a$, $a^{11}+a^{10}+10a^{9}+10a^{8}+35a^{7}+35a^{6}+49a^{5}+49a^{4}+22a^{3}+22a^{2}+3a+2$, $a^{15}+16a^{13}+a^{12}+104a^{11}+12a^{10}+351a^{9}+54a^{8}+650a^{7}+111a^{6}+637a^{5}+98a^{4}+286a^{3}+23a^{2}+39a-2$, $a^{15}+16a^{13}-a^{12}+104a^{11}-12a^{10}+350a^{9}-54a^{8}+641a^{7}-112a^{6}+611a^{5}-105a^{4}+262a^{3}-37a^{2}+39a-4$, $a^{16}-a^{15}+16a^{14}-15a^{13}+104a^{12}-90a^{11}+351a^{10}-274a^{9}+651a^{8}-441a^{7}+645a^{6}-351a^{5}+308a^{4}-111a^{3}+62a^{2}-9a+5$, $a^{16}+a^{15}+17a^{14}+17a^{13}+118a^{12}+119a^{11}+429a^{10}+441a^{9}+869a^{8}+925a^{7}+957a^{6}+1085a^{5}+504a^{4}+651a^{3}+79a^{2}+154a-8$, $a^{15}+a^{14}+13a^{13}+13a^{12}+66a^{11}+66a^{10}+165a^{9}+165a^{8}+209a^{7}+209a^{6}+121a^{5}+121a^{4}+21a^{3}+21a^{2}-a-2$, $2a^{15}+29a^{13}+a^{12}+166a^{11}+12a^{10}+475a^{9}+53a^{8}+710a^{7}+104a^{6}+529a^{5}+87a^{4}+169a^{3}+27a^{2}+16a+1$ | 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: | \( 7413165.64743 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 7413165.64743 \cdot 1}{2\cdot\sqrt{323140727299350298506640625}}\cr\approx \mathstrut & 1.00172105530 \end{aligned}\]
Galois group
A solvable group of order 272 |
The 17 conjugacy class representatives for $F_{17}$ |
Character table for $F_{17}$ |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | $16{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.16.8.2 | $x^{16} + 625 x^{8} + 46875 x^{4} - 312500 x^{2} + 781250$ | $2$ | $8$ | $8$ | $C_{16}$ | $[\ ]_{2}^{8}$ | |
\(17\) | 17.17.17.8 | $x^{17} + 272 x + 17$ | $17$ | $1$ | $17$ | $F_{17}$ | $[17/16]_{16}$ |