Normalized defining polynomial
\( x^{12} - 3x^{11} + 2x^{10} + x^{9} + 4x^{8} - 5x^{7} - 20x^{6} + 28x^{5} - 21x^{4} + 35x^{3} - 21x^{2} + 7x - 7 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(8067775586689\) \(\medspace = 7^{10}\cdot 13^{4}\) | 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: | \(11.90\) | 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: | $7^{5/6}13^{1/2}\approx 18.248200448594663$ | ||
Ramified primes: | \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{3}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{8}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{9}-\frac{2}{5}a^{6}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{8425}a^{11}-\frac{354}{8425}a^{10}-\frac{434}{8425}a^{9}+\frac{137}{1685}a^{8}+\frac{524}{8425}a^{7}-\frac{3634}{8425}a^{6}+\frac{1654}{8425}a^{5}+\frac{2484}{8425}a^{4}-\frac{152}{1685}a^{3}-\frac{224}{1685}a^{2}-\frac{2876}{8425}a-\frac{3202}{8425}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{23}{8425}a^{11}+\frac{283}{8425}a^{10}-\frac{1557}{8425}a^{9}+\frac{91}{337}a^{8}+\frac{257}{8425}a^{7}-\frac{1017}{8425}a^{6}-\frac{4083}{8425}a^{5}-\frac{3528}{8425}a^{4}+\frac{4929}{1685}a^{3}-\frac{963}{337}a^{2}+\frac{6307}{8425}a-\frac{6246}{8425}$, $\frac{1228}{8425}a^{11}-\frac{1667}{8425}a^{10}-\frac{2177}{8425}a^{9}+\frac{82}{337}a^{8}+\frac{8227}{8425}a^{7}+\frac{2698}{8425}a^{6}-\frac{29643}{8425}a^{5}-\frac{6238}{8425}a^{4}+\frac{1053}{1685}a^{3}+\frac{3964}{1685}a^{2}+\frac{10142}{8425}a-\frac{4321}{8425}$, $\frac{1228}{8425}a^{11}-\frac{1667}{8425}a^{10}-\frac{2177}{8425}a^{9}+\frac{82}{337}a^{8}+\frac{8227}{8425}a^{7}+\frac{2698}{8425}a^{6}-\frac{29643}{8425}a^{5}-\frac{6238}{8425}a^{4}+\frac{1053}{1685}a^{3}+\frac{3964}{1685}a^{2}+\frac{10142}{8425}a+\frac{4104}{8425}$, $\frac{134}{1685}a^{11}-\frac{593}{1685}a^{10}+\frac{819}{1685}a^{9}-\frac{211}{1685}a^{8}+\frac{24}{337}a^{7}-\frac{1002}{1685}a^{6}-\frac{359}{337}a^{5}+\frac{6977}{1685}a^{4}-\frac{8491}{1685}a^{3}+\frac{1662}{337}a^{2}-\frac{5248}{1685}a+\frac{1618}{1685}$, $\frac{878}{8425}a^{11}-\frac{2457}{8425}a^{10}-\frac{242}{8425}a^{9}+\frac{988}{1685}a^{8}+\frac{3437}{8425}a^{7}-\frac{7687}{8425}a^{6}-\frac{25533}{8425}a^{5}+\frac{27522}{8425}a^{4}+\frac{4377}{1685}a^{3}-\frac{201}{1685}a^{2}-\frac{12793}{8425}a-\frac{22681}{8425}$, $\frac{638}{1685}a^{11}-\frac{282}{337}a^{10}+\frac{459}{1685}a^{9}+\frac{278}{1685}a^{8}+\frac{3041}{1685}a^{7}-\frac{269}{1685}a^{6}-\frac{12027}{1685}a^{5}+\frac{7632}{1685}a^{4}-\frac{12743}{1685}a^{3}+\frac{18078}{1685}a^{2}-\frac{726}{337}a+\frac{6084}{1685}$, $\frac{72}{1685}a^{11}+\frac{124}{1685}a^{10}-\frac{581}{1685}a^{9}+\frac{118}{1685}a^{8}+\frac{199}{337}a^{7}+\frac{175}{337}a^{6}-\frac{379}{337}a^{5}-\frac{896}{337}a^{4}+\frac{3244}{1685}a^{3}-\frac{289}{337}a^{2}+\frac{1531}{1685}a+\frac{1649}{1685}$ | 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: | \( 51.9911514065 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 51.9911514065 \cdot 1}{2\cdot\sqrt{8067775586689}}\cr\approx \mathstrut & 0.228224356518 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T6):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4\times C_2$ |
Character table for $A_4\times C_2$ |
Intermediate fields
\(\Q(\zeta_{7})^+\), 6.4.218491.1 x2, 6.2.405769.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.4.218491.1 |
Degree 8 sibling: | 8.0.3360173089.1 |
Degree 12 sibling: | 12.0.1363454074150441.3 |
Minimal sibling: | 6.4.218491.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.6.5.2 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
7.6.5.2 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |