Normalized defining polynomial
\( x^{12} - 15x^{10} + 76x^{8} - 148x^{6} + 99x^{4} - 19x^{2} + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(52206766144000000\) \(\medspace = 2^{12}\cdot 5^{6}\cdot 13^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.73\) | 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: | $2^{3/2}5^{1/2}13^{2/3}\approx 34.96704216279658$ | ||
Ramified primes: | \(2\), \(5\), \(13\) | 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)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}-\frac{1}{5}a^{6}-\frac{2}{5}a^{4}-\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{7}-\frac{2}{5}a^{5}-\frac{2}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{5}a^{10}+\frac{2}{5}a^{6}+\frac{1}{5}a^{4}+\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{11}+\frac{2}{5}a^{7}+\frac{1}{5}a^{5}+\frac{2}{5}a^{3}-\frac{1}{5}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{3}{5}a^{10}-\frac{44}{5}a^{8}+43a^{6}-\frac{389}{5}a^{4}+\frac{214}{5}a^{2}-\frac{19}{5}$, $\frac{4}{5}a^{11}-12a^{9}+\frac{303}{5}a^{7}-\frac{581}{5}a^{5}+\frac{358}{5}a^{3}-\frac{34}{5}a$, $a$, $\frac{3}{5}a^{10}-9a^{8}+\frac{226}{5}a^{6}-\frac{427}{5}a^{4}+\frac{261}{5}a^{2}-\frac{33}{5}$, $\frac{7}{5}a^{11}+\frac{1}{5}a^{10}-\frac{104}{5}a^{9}-3a^{8}+\frac{518}{5}a^{7}+\frac{77}{5}a^{6}-194a^{5}-\frac{154}{5}a^{4}+\frac{572}{5}a^{3}+\frac{97}{5}a^{2}-\frac{53}{5}a-\frac{6}{5}$, $\frac{12}{5}a^{11}-\frac{3}{5}a^{10}-\frac{178}{5}a^{9}+\frac{44}{5}a^{8}+\frac{882}{5}a^{7}-43a^{6}-\frac{1627}{5}a^{5}+\frac{389}{5}a^{4}+184a^{3}-\frac{214}{5}a^{2}-\frac{94}{5}a+\frac{24}{5}$, $\frac{3}{5}a^{11}+\frac{1}{5}a^{10}-\frac{44}{5}a^{9}-3a^{8}+43a^{7}+\frac{77}{5}a^{6}-\frac{389}{5}a^{5}-\frac{154}{5}a^{4}+\frac{214}{5}a^{3}+\frac{102}{5}a^{2}-\frac{19}{5}a-\frac{16}{5}$, $\frac{16}{5}a^{11}+\frac{4}{5}a^{10}-\frac{239}{5}a^{9}-12a^{8}+\frac{1201}{5}a^{7}+\frac{303}{5}a^{6}-\frac{2291}{5}a^{5}-\frac{581}{5}a^{4}+286a^{3}+\frac{363}{5}a^{2}-\frac{202}{5}a-\frac{54}{5}$, $\frac{9}{5}a^{11}-\frac{2}{5}a^{10}-\frac{134}{5}a^{9}+\frac{31}{5}a^{8}+\frac{667}{5}a^{7}-32a^{6}-\frac{1238}{5}a^{5}+\frac{306}{5}a^{4}+\frac{706}{5}a^{3}-\frac{181}{5}a^{2}-13a+\frac{16}{5}$, $\frac{9}{5}a^{11}+\frac{2}{5}a^{10}-\frac{134}{5}a^{9}-\frac{31}{5}a^{8}+\frac{667}{5}a^{7}+32a^{6}-\frac{1238}{5}a^{5}-\frac{306}{5}a^{4}+\frac{706}{5}a^{3}+\frac{181}{5}a^{2}-13a-\frac{16}{5}$, $\frac{4}{5}a^{11}-\frac{3}{5}a^{10}-12a^{9}+9a^{8}+\frac{303}{5}a^{7}-\frac{226}{5}a^{6}-\frac{581}{5}a^{5}+\frac{422}{5}a^{4}+\frac{363}{5}a^{3}-\frac{231}{5}a^{2}-\frac{59}{5}a+\frac{23}{5}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 27689.0510913 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 27689.0510913 \cdot 1}{2\cdot\sqrt{52206766144000000}}\cr\approx \mathstrut & 0.248184485115 \end{aligned}\] (assuming GRH)
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
\(\Q(\sqrt{5}) \), 3.3.169.1, 6.6.9139520.1, 6.6.3570125.1, 6.6.45697600.1 |
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: | 6.6.9139520.1 |
Degree 8 sibling: | 8.8.73116160000.1 |
Degree 12 sibling: | deg 12 |
Minimal sibling: | 6.6.9139520.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.11 | $x^{12} + 28 x^{10} + 40 x^{9} + 356 x^{8} + 896 x^{7} + 2720 x^{6} + 6656 x^{5} + 12464 x^{4} + 19456 x^{3} + 26304 x^{2} + 19840 x + 5824$ | $2$ | $6$ | $12$ | $A_4 \times C_2$ | $[2, 2]^{6}$ |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |