Normalized defining polynomial
\( x^{8} - 4x^{7} + 12x^{6} - 22x^{5} + 48x^{4} - 64x^{3} + 89x^{2} - 60x + 58 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(796594176\) \(\medspace = 2^{12}\cdot 3^{4}\cdot 7^{4}\) | 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: | \(12.96\) | 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: | $2^{3/2}3^{1/2}7^{1/2}\approx 12.96148139681572$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | 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)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{17}a^{6}-\frac{3}{17}a^{5}-\frac{1}{17}a^{4}+\frac{7}{17}a^{3}-\frac{3}{17}a^{2}-\frac{1}{17}a-\frac{1}{17}$, $\frac{1}{1819}a^{7}+\frac{50}{1819}a^{6}-\frac{177}{1819}a^{5}+\frac{906}{1819}a^{4}-\frac{890}{1819}a^{3}+\frac{775}{1819}a^{2}-\frac{326}{1819}a-\frac{223}{1819}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | 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: | $\frac{70}{1819}a^{7}-\frac{352}{1819}a^{6}+\frac{985}{1819}a^{5}-\frac{1850}{1819}a^{4}+\frac{3505}{1819}a^{3}-\frac{5135}{1819}a^{2}+\frac{4679}{1819}a-\frac{2663}{1819}$, $\frac{150}{1819}a^{7}-\frac{311}{1819}a^{6}+\frac{521}{1819}a^{5}+\frac{10}{1819}a^{4}+\frac{999}{1819}a^{3}+\frac{1439}{1819}a^{2}-\frac{170}{107}a+\frac{7103}{1819}$, $\frac{6}{17}a^{6}-\frac{18}{17}a^{5}+\frac{45}{17}a^{4}-\frac{60}{17}a^{3}+\frac{152}{17}a^{2}-\frac{125}{17}a+\frac{147}{17}$ | 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: | \( 23.076993704 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 23.076993704 \cdot 2}{2\cdot\sqrt{796594176}}\cr\approx \mathstrut & 1.2743248189 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 8T9):
A solvable group of order 16 |
The 10 conjugacy class representatives for $D_4\times C_2$ |
Character table for $D_4\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{42}) \), 4.0.3528.1, 4.0.392.1, \(\Q(\sqrt{-6}, \sqrt{-7})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 16.0.634562281237118976.5 |
Degree 8 siblings: | 8.0.16257024.1, 8.0.12446784.1, 8.4.796594176.2 |
Minimal sibling: | 8.0.12446784.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(7\) | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |