Normalized defining polynomial
\( x^{8} - 2x^{7} + 51x^{6} + 16x^{5} - 1409x^{4} + 3618x^{3} - 35251x^{2} - 129244x + 1048631 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(106837547250625\) \(\medspace = 5^{4}\cdot 643^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(56.70\) | 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: | $5^{1/2}643^{1/2}\approx 56.70097000933934$ | ||
Ramified primes: | \(5\), \(643\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{74\!\cdots\!14}a^{7}+\frac{763294146835823}{74\!\cdots\!14}a^{6}+\frac{12\!\cdots\!67}{74\!\cdots\!14}a^{5}-\frac{129997295730519}{74\!\cdots\!14}a^{4}-\frac{826343920726973}{74\!\cdots\!14}a^{3}+\frac{565270584825377}{37\!\cdots\!07}a^{2}-\frac{344777135831268}{37\!\cdots\!07}a+\frac{17\!\cdots\!92}{37\!\cdots\!07}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{50}$, which has order $200$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{28632970983}{37\!\cdots\!07}a^{7}+\frac{219796661681}{74\!\cdots\!14}a^{6}+\frac{1509909628296}{37\!\cdots\!07}a^{5}+\frac{13001219680596}{37\!\cdots\!07}a^{4}-\frac{14314743491684}{37\!\cdots\!07}a^{3}+\frac{58878871171873}{37\!\cdots\!07}a^{2}-\frac{628735156131437}{74\!\cdots\!14}a-\frac{95\!\cdots\!59}{37\!\cdots\!07}$, $\frac{29213579093}{74\!\cdots\!14}a^{7}-\frac{549119684}{37\!\cdots\!07}a^{6}+\frac{964367601906}{37\!\cdots\!07}a^{5}+\frac{6575264928325}{37\!\cdots\!07}a^{4}-\frac{1334848960560}{37\!\cdots\!07}a^{3}+\frac{538232325889891}{74\!\cdots\!14}a^{2}-\frac{246295542369046}{37\!\cdots\!07}a-\frac{34\!\cdots\!14}{37\!\cdots\!07}$, $\frac{1146812609619}{74\!\cdots\!14}a^{7}-\frac{8027492574317}{74\!\cdots\!14}a^{6}+\frac{96653625299013}{74\!\cdots\!14}a^{5}-\frac{447948985837937}{74\!\cdots\!14}a^{4}+\frac{411790185548225}{74\!\cdots\!14}a^{3}+\frac{17\!\cdots\!84}{37\!\cdots\!07}a^{2}-\frac{32\!\cdots\!15}{37\!\cdots\!07}a+\frac{11\!\cdots\!22}{37\!\cdots\!07}$ | 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: | \( 26.1311328719 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 26.1311328719 \cdot 200}{2\cdot\sqrt{106837547250625}}\cr\approx \mathstrut & 0.394017723220 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 8T24):
A solvable group of order 48 |
The 10 conjugacy class representatives for $S_4\times C_2$ |
Character table for $S_4\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-3215}) \), 4.2.643.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.0.33230963375.2, 6.2.51681125.1 |
Degree 8 sibling: | 8.4.258405625.2 |
Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 |
Degree 16 sibling: | deg 16 |
Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
Minimal sibling: | 6.2.51681125.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(643\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |