Normalized defining polynomial
\( x^{8} - 3x^{7} - 67x^{6} - 16x^{5} + 863x^{4} + 1276x^{3} + 392x^{2} - 54x + 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(116507435287321\) \(\medspace = 13^{6}\cdot 17^{6}\) | 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: | \(57.32\) | 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: | $13^{3/4}17^{3/4}\approx 57.318419318377835$ | ||
Ramified primes: | \(13\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{8232013}a^{7}+\frac{3019361}{8232013}a^{6}-\frac{1122526}{8232013}a^{5}+\frac{2726932}{8232013}a^{4}-\frac{1466398}{8232013}a^{3}-\frac{1137546}{8232013}a^{2}+\frac{2039677}{8232013}a+\frac{1971827}{8232013}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $7$ | 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: | $\frac{218396}{8232013}a^{7}-\frac{804396}{8232013}a^{6}-\frac{14073169}{8232013}a^{5}+\frac{6060587}{8232013}a^{4}+\frac{183880430}{8232013}a^{3}+\frac{154832358}{8232013}a^{2}-\frac{18085403}{8232013}a-\frac{2166577}{8232013}$, $\frac{885973}{8232013}a^{7}-\frac{2621227}{8232013}a^{6}-\frac{59397333}{8232013}a^{5}-\frac{17138521}{8232013}a^{4}+\frac{761065628}{8232013}a^{3}+\frac{1171023165}{8232013}a^{2}+\frac{427089624}{8232013}a-\frac{12316189}{8232013}$, $\frac{43328}{8232013}a^{7}-\frac{277188}{8232013}a^{6}-\frac{2073724}{8232013}a^{5}+\frac{6659120}{8232013}a^{4}+\frac{23279829}{8232013}a^{3}-\frac{27227296}{8232013}a^{2}-\frac{11998525}{8232013}a+\frac{11721355}{8232013}$, $\frac{2711054}{8232013}a^{7}-\frac{7818303}{8232013}a^{6}-\frac{182676824}{8232013}a^{5}-\frac{64008582}{8232013}a^{4}+\frac{2339766290}{8232013}a^{3}+\frac{3723034569}{8232013}a^{2}+\frac{1400303304}{8232013}a-\frac{32750373}{8232013}$, $\frac{885973}{8232013}a^{7}-\frac{2621227}{8232013}a^{6}-\frac{59397333}{8232013}a^{5}-\frac{17138521}{8232013}a^{4}+\frac{761065628}{8232013}a^{3}+\frac{1171023165}{8232013}a^{2}+\frac{418857611}{8232013}a-\frac{20548202}{8232013}$, $\frac{149208}{8232013}a^{7}-\frac{559363}{8232013}a^{6}-\frac{9554923}{8232013}a^{5}+\frac{4595318}{8232013}a^{4}+\frac{123840938}{8232013}a^{3}+\frac{103696635}{8232013}a^{2}+\frac{15069232}{8232013}a+\frac{218396}{8232013}$, $\frac{1806}{6731}a^{7}-\frac{5602}{6731}a^{6}-\frac{120148}{6731}a^{5}-\frac{18116}{6731}a^{4}+\frac{1545292}{6731}a^{3}+\frac{2168202}{6731}a^{2}+\frac{668854}{6731}a-\frac{16953}{6731}$ | 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: | \( 7915.1159478 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 7915.1159478 \cdot 2}{2\cdot\sqrt{116507435287321}}\cr\approx \mathstrut & 0.18772427055 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $Q_8$ |
Character table for $Q_8$ |
Intermediate fields
\(\Q(\sqrt{221}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{17})\) |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.4.3.1 | $x^{4} + 52$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
13.4.3.1 | $x^{4} + 52$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(17\) | 17.4.3.2 | $x^{4} + 34$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
17.4.3.2 | $x^{4} + 34$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.221.2t1.a.a | $1$ | $ 13 \cdot 17 $ | \(\Q(\sqrt{221}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.48841.8t5.a.a | $2$ | $ 13^{2} \cdot 17^{2}$ | 8.8.116507435287321.1 | $Q_8$ (as 8T5) | $-1$ | $2$ |