Normalized defining polynomial
\( x^{8} - 7x^{6} - 3x^{5} - 54x^{4} + 147x^{3} + 473x^{2} - 1095x + 547 \)
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: | \(49991146569\) \(\medspace = 3^{6}\cdot 7^{4}\cdot 13^{4}\) | 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: | \(21.75\) | 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: | $3^{3/4}7^{1/2}13^{1/2}\approx 21.745111415357325$ | ||
Ramified primes: | \(3\), \(7\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{6}a^{4}-\frac{1}{6}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{26253738}a^{7}-\frac{920308}{13126869}a^{6}+\frac{98394}{1458541}a^{5}+\frac{249815}{1458541}a^{4}+\frac{3351259}{8751246}a^{3}-\frac{412522}{1458541}a^{2}-\frac{8263567}{26253738}a-\frac{5717893}{26253738}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{8551}{625089} a^{7} - \frac{8515}{625089} a^{6} + \frac{16200}{208363} a^{5} + \frac{24829}{208363} a^{4} + \frac{172770}{208363} a^{3} - \frac{243082}{208363} a^{2} - \frac{4550033}{625089} a + \frac{4867264}{625089} \) (order $6$) | 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: | $\frac{561373}{13126869}a^{7}+\frac{616321}{13126869}a^{6}-\frac{1110412}{4375623}a^{5}-\frac{1761094}{4375623}a^{4}-\frac{3917092}{1458541}a^{3}+\frac{14100119}{4375623}a^{2}+\frac{310888259}{13126869}a-\frac{268139752}{13126869}$, $\frac{262465}{3750534}a^{7}+\frac{127427}{1875267}a^{6}-\frac{253934}{625089}a^{5}-\frac{341321}{625089}a^{4}-\frac{1792063}{416726}a^{3}+\frac{3696133}{625089}a^{2}+\frac{139214141}{3750534}a-\frac{163133557}{3750534}$, $\frac{17915}{2917082}a^{7}+\frac{29644}{1458541}a^{6}+\frac{6133}{1458541}a^{5}-\frac{146731}{1458541}a^{4}-\frac{1813135}{2917082}a^{3}-\frac{597988}{1458541}a^{2}+\frac{5942859}{2917082}a-\frac{2718665}{2917082}$ | 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: | \( 189.845040565 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 189.845040565 \cdot 4}{6\cdot\sqrt{49991146569}}\cr\approx \mathstrut & 0.882228171942 \end{aligned}\]
Galois group
$C_2^2\wr C_2$ (as 8T18):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_2^2 \wr C_2$ |
Character table for $C_2^2 \wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.2457.2, 4.0.31941.1, 4.0.5733.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 8 siblings: | data not computed |
Degree 16 siblings: | 16.0.1040864112987605361.1, 16.0.87500953582971201.1, 16.0.2499114735283240471761.1, 16.0.2499114735283240471761.5, 16.0.2499114735283240471761.2, 16.0.2499114735283240471761.3, 16.4.2499114735283240471761.1 |
Minimal sibling: | 8.0.6036849.2 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |