Normalized defining polynomial
\( x^{12} - 4x^{10} - x^{8} - x^{4} - 4x^{2} + 1 \)
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: | \(96717311574016\) \(\medspace = 2^{24}\cdot 7^{8}\) | 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: | \(14.64\) | 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: | $2^{9/4}7^{2/3}\approx 17.406689545317608$ | ||
Ramified primes: | \(2\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{3}$
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: | $\frac{1}{4}a^{10}-a^{8}-a^{4}-\frac{1}{4}a^{2}-1$, $a$, $\frac{1}{4}a^{8}-a^{6}-a^{2}-\frac{1}{4}$, $\frac{3}{4}a^{11}-\frac{11}{4}a^{9}-\frac{3}{2}a^{7}-\frac{3}{2}a^{5}-\frac{1}{4}a^{3}-\frac{11}{4}a+1$, $\frac{3}{4}a^{11}-\frac{11}{4}a^{9}-\frac{1}{4}a^{8}-\frac{3}{2}a^{7}+a^{6}-\frac{3}{2}a^{5}+\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-a^{2}-\frac{11}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{3}{4}a^{9}-\frac{3}{4}a^{8}-a^{7}-a^{6}-a^{5}-a^{4}-\frac{5}{4}a^{3}-\frac{5}{4}a^{2}-\frac{9}{4}a-\frac{5}{4}$, $a^{11}-\frac{1}{2}a^{10}-\frac{7}{2}a^{9}+\frac{7}{4}a^{8}-3a^{7}+\frac{3}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{2}a^{4}-a^{3}-4a+\frac{7}{4}$ | 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: | \( 245.785554293 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 245.785554293 \cdot 1}{2\cdot\sqrt{96717311574016}}\cr\approx \mathstrut & 0.311611664840 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T7):
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
\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), 6.2.1229312.1, 6.6.1229312.1, 6.2.153664.1 |
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: | 6.2.1229312.1 |
Degree 8 sibling: | 8.0.629407744.4 |
Degree 12 sibling: | 12.0.96717311574016.5 |
Minimal sibling: | 6.2.1229312.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.24.123 | $x^{12} + 6 x^{10} + 4 x^{9} + 50 x^{8} + 136 x^{7} + 224 x^{6} + 288 x^{5} + 140 x^{4} + 592 x^{3} + 664 x^{2} + 1776 x + 632$ | $4$ | $3$ | $24$ | $A_4 \times C_2$ | $[2, 2, 3]^{3}$ |
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |