Normalized defining polynomial
\( x^{8} - 56x^{6} - 34x^{5} + 1007x^{4} + 952x^{3} - 5955x^{2} - 3791x + 11943 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(14667743362801\) \(\medspace = 19^{4}\cdot 103^{4}\) | 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: | \(44.24\) | 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: | $19^{1/2}103^{1/2}\approx 44.23799272118933$ | ||
Ramified primes: | \(19\), \(103\) | 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)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{9768443}a^{7}+\frac{3601766}{9768443}a^{6}+\frac{1340511}{9768443}a^{5}-\frac{2305446}{9768443}a^{4}-\frac{2276036}{9768443}a^{3}-\frac{3333923}{9768443}a^{2}+\frac{4338865}{9768443}a+\frac{1783513}{9768443}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
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: | $\frac{65232}{9768443}a^{7}-\frac{191324}{9768443}a^{6}-\frac{2888184}{9768443}a^{5}+\frac{6094956}{9768443}a^{4}+\frac{39258577}{9768443}a^{3}-\frac{52460842}{9768443}a^{2}-\frac{134784004}{9768443}a+\frac{185564303}{9768443}$, $\frac{171992}{9768443}a^{7}-\frac{643416}{9768443}a^{6}-\frac{7392217}{9768443}a^{5}+\frac{21906710}{9768443}a^{4}+\frac{98285500}{9768443}a^{3}-\frac{205617819}{9768443}a^{2}-\frac{342260967}{9768443}a+\frac{636269605}{9768443}$, $\frac{106760}{9768443}a^{7}-\frac{452092}{9768443}a^{6}-\frac{4504033}{9768443}a^{5}+\frac{15811754}{9768443}a^{4}+\frac{59026923}{9768443}a^{3}-\frac{153156977}{9768443}a^{2}-\frac{207476963}{9768443}a+\frac{450705302}{9768443}$, $\frac{747152}{9768443}a^{7}-\frac{2617866}{9768443}a^{6}-\frac{32059890}{9768443}a^{5}+\frac{87722600}{9768443}a^{4}+\frac{424361675}{9768443}a^{3}-\frac{805053065}{9768443}a^{2}-\frac{1477940165}{9768443}a+\frac{2454800767}{9768443}$, $\frac{8139}{9768443}a^{7}-\frac{323969}{9768443}a^{6}-\frac{931802}{9768443}a^{5}+\frac{10922452}{9768443}a^{4}+\frac{25616253}{9768443}a^{3}-\frac{76212187}{9768443}a^{2}-\frac{106352083}{9768443}a+\frac{175937983}{9768443}$, $\frac{867356}{9768443}a^{7}-\frac{3099805}{9768443}a^{6}-\frac{37511374}{9768443}a^{5}+\frac{104387969}{9768443}a^{4}+\frac{498660976}{9768443}a^{3}-\frac{952085927}{9768443}a^{2}-\frac{1690857664}{9768443}a+\frac{2794074403}{9768443}$, $\frac{142880}{9768443}a^{7}-\frac{788046}{9768443}a^{6}-\frac{7418664}{9768443}a^{5}+\frac{19078809}{9768443}a^{4}+\frac{128201992}{9768443}a^{3}-\frac{31869117}{9768443}a^{2}-\frac{447015287}{9768443}a-\frac{69414202}{9768443}$ | 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: | \( 6402.557564 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 6402.557564 \cdot 3}{2\cdot\sqrt{14667743362801}}\cr\approx \mathstrut & 0.6419527518 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
\(\Q(\sqrt{1957}) \), 4.4.1957.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 4 sibling: | 4.4.1957.1 |
Degree 6 siblings: | 6.6.3829849.1, 6.6.7495014493.2 |
Degree 12 siblings: | 12.12.28704773761001557.1, deg 12 |
Minimal sibling: | 4.4.1957.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(19\) | 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(103\) | 103.4.2.1 | $x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
103.4.2.1 | $x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |