Normalized defining polynomial
\( x^{12} - x^{11} + 2x^{10} + 7x^{9} - x^{8} + 9x^{6} - x^{4} + 7x^{3} + 2x^{2} - x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(371667309755625\) \(\medspace = 3^{6}\cdot 5^{4}\cdot 13^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.37\) | 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: | $3^{1/2}5^{1/2}13^{2/3}\approx 21.412852778309286$ | ||
Ramified primes: | \(3\), \(5\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{10}+\frac{1}{5}a^{8}+\frac{1}{5}a^{6}-\frac{1}{5}a^{4}-\frac{1}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{11}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{2}{5} a^{11} - \frac{22}{5} a^{8} - \frac{3}{5} a^{7} + \frac{6}{5} a^{6} - \frac{22}{5} a^{5} - \frac{6}{5} a^{4} + \frac{6}{5} a^{3} - \frac{14}{5} a^{2} - \frac{6}{5} a + \frac{3}{5} \) (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: | $\frac{2}{5}a^{11}-\frac{4}{5}a^{10}+\frac{8}{5}a^{9}+a^{8}-\frac{6}{5}a^{7}+\frac{4}{5}a^{6}+\frac{6}{5}a^{5}-\frac{4}{5}a^{4}+\frac{2}{5}a^{2}+\frac{4}{5}a-\frac{2}{5}$, $a^{11}-\frac{1}{5}a^{10}+\frac{6}{5}a^{9}+\frac{43}{5}a^{8}+\frac{22}{5}a^{7}-\frac{3}{5}a^{6}+\frac{43}{5}a^{5}+\frac{28}{5}a^{4}-\frac{3}{5}a^{3}+\frac{29}{5}a^{2}+\frac{26}{5}a$, $\frac{2}{5}a^{11}-\frac{4}{5}a^{10}+\frac{8}{5}a^{9}+a^{8}-\frac{6}{5}a^{7}+\frac{4}{5}a^{6}+\frac{6}{5}a^{5}-\frac{4}{5}a^{4}+\frac{2}{5}a^{2}+\frac{4}{5}a-\frac{7}{5}$, $\frac{7}{5}a^{11}-\frac{9}{5}a^{10}+\frac{18}{5}a^{9}+8a^{8}-\frac{11}{5}a^{7}+\frac{4}{5}a^{6}+\frac{51}{5}a^{5}-\frac{4}{5}a^{4}-a^{3}+\frac{37}{5}a^{2}+\frac{14}{5}a-\frac{7}{5}$, $\frac{2}{5}a^{11}-\frac{2}{5}a^{10}+a^{9}+3a^{8}-\frac{2}{5}a^{7}+\frac{12}{5}a^{6}+\frac{27}{5}a^{5}+\frac{3}{5}a^{4}+\frac{4}{5}a^{3}+\frac{16}{5}a^{2}-\frac{4}{5}a-\frac{6}{5}$ | 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: | \( 265.862081831 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 265.862081831 \cdot 2}{6\cdot\sqrt{371667309755625}}\cr\approx \mathstrut & 0.282837620273 \end{aligned}\]
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
\(\Q(\sqrt{-3}) \), 3.3.169.1, 6.4.19278675.1, 6.0.771147.1, 6.2.714025.1 |
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: | 6.4.19278675.1 |
Degree 8 sibling: | 8.0.1445900625.6 |
Degree 12 sibling: | 12.4.9291682743890625.1 |
Minimal sibling: | 6.4.19278675.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | R | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |