Normalized defining polynomial
\( x^{8} - 2x^{7} - 14x^{6} + 14x^{5} + 28x^{4} + 70x^{3} + 70x^{2} - 2x - 15 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1129509830656\) \(\medspace = 2^{14}\cdot 19^{4}\cdot 23^{2}\) | 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: | \(32.11\) | 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: | $2^{7/4}19^{1/2}23^{1/2}\approx 70.31422767873043$ | ||
Ramified primes: | \(2\), \(19\), \(23\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{115}a^{6}-\frac{28}{115}a^{5}+\frac{16}{115}a^{4}+\frac{52}{115}a^{3}+\frac{43}{115}a^{2}+\frac{31}{115}a-\frac{9}{23}$, $\frac{1}{115}a^{7}+\frac{37}{115}a^{5}+\frac{8}{23}a^{4}+\frac{4}{115}a^{3}-\frac{6}{23}a^{2}+\frac{18}{115}a+\frac{1}{23}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | 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: | $\frac{13}{23}a^{7}-\frac{173}{115}a^{6}-\frac{801}{115}a^{5}+\frac{1442}{115}a^{4}+\frac{924}{115}a^{3}+\frac{3951}{115}a^{2}+\frac{2017}{115}a-\frac{310}{23}$, $\frac{11}{115}a^{7}-\frac{8}{115}a^{6}-\frac{174}{115}a^{5}-\frac{33}{115}a^{4}+\frac{433}{115}a^{3}+\frac{936}{115}a^{2}+\frac{312}{23}a+\frac{152}{23}$, $\frac{4}{115}a^{6}+\frac{3}{115}a^{5}-\frac{51}{115}a^{4}-\frac{22}{115}a^{3}-\frac{58}{115}a^{2}+\frac{9}{115}a+\frac{10}{23}$, $\frac{3}{23}a^{7}-\frac{44}{115}a^{6}-\frac{168}{115}a^{5}+\frac{356}{115}a^{4}+\frac{72}{115}a^{3}+\frac{1108}{115}a^{2}+\frac{171}{115}a-\frac{49}{23}$, $\frac{1}{115}a^{7}+\frac{2}{23}a^{6}-\frac{13}{115}a^{5}-\frac{29}{23}a^{4}-\frac{51}{115}a^{3}+\frac{34}{23}a^{2}-\frac{17}{115}a-\frac{20}{23}$ | 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: | \( 3162.95322615 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 3162.95322615 \cdot 2}{2\cdot\sqrt{1129509830656}}\cr\approx \mathstrut & 1.87986854638 \end{aligned}\]
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
\(\Q(\sqrt{19}) \), 4.2.531392.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.643264.2, 6.0.14795072.2 |
Degree 8 sibling: | 8.0.597510700417024.6 |
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: | 6.2.643264.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.14.1 | $x^{8} + 2 x^{7} + 6$ | $8$ | $1$ | $14$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ |
\(19\) | 19.8.4.1 | $x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(23\) | 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |