Normalized defining polynomial
\( x^{8} - 3x^{7} + 12x^{5} - 25x^{4} + 24x^{3} - 24x + 16 \)
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: | \(390971529\) \(\medspace = 3^{4}\cdot 13^{6}\) | 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: | \(11.86\) | 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: | $3^{1/2}13^{3/4}\approx 11.858182817915045$ | ||
Ramified primes: | \(3\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{7}{8}a^{7}-\frac{13}{8}a^{6}-2a^{5}+\frac{17}{2}a^{4}-\frac{95}{8}a^{3}+6a^{2}+9a-12$, $\frac{7}{8}a^{7}-\frac{13}{8}a^{6}-2a^{5}+\frac{17}{2}a^{4}-\frac{95}{8}a^{3}+6a^{2}+9a-11$, $a-1$, $\frac{7}{8}a^{7}-\frac{9}{8}a^{6}-2a^{5}+7a^{4}-\frac{79}{8}a^{3}+\frac{9}{2}a^{2}+\frac{17}{2}a-9$, $\frac{5}{8}a^{7}-\frac{5}{8}a^{6}-\frac{7}{4}a^{5}+\frac{9}{2}a^{4}-\frac{45}{8}a^{3}+\frac{3}{4}a^{2}+7a-5$ | 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: | \( 17.3591776333 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 17.3591776333 \cdot 1}{2\cdot\sqrt{390971529}}\cr\approx \mathstrut & 0.277272184848 \end{aligned}\]
Galois group
$C_2^2:C_4$ (as 8T10):
A solvable group of order 16 |
The 10 conjugacy class representatives for $C_2^2:C_4$ |
Character table for $C_2^2:C_4$ |
Intermediate fields
\(\Q(\sqrt{13}) \), 4.4.19773.1, 4.2.6591.1, 4.2.507.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 16.0.152858736488597841.1 |
Degree 8 sibling: | 8.0.43441281.1 |
Minimal sibling: | 8.0.43441281.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.8.6.1 | $x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24242 x^{4} + 15024 x^{3} + 14408 x^{2} + 86496 x + 254881$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |