Normalized defining polynomial
\( x^{12} - 5x^{10} + 18x^{8} - 20x^{6} - 7x^{4} - 15x^{2} + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(86161557405625\) \(\medspace = 5^{4}\cdot 13^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.50\) | 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: | $5^{1/2}13^{5/6}\approx 18.957066304919827$ | ||
Ramified primes: | \(5\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{30}a^{8}-\frac{1}{2}a^{5}-\frac{13}{30}a^{4}+\frac{1}{6}a^{2}+\frac{1}{30}$, $\frac{1}{30}a^{9}+\frac{1}{15}a^{5}+\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{7}{15}a-\frac{1}{2}$, $\frac{1}{90}a^{10}-\frac{1}{90}a^{8}-\frac{13}{90}a^{6}+\frac{1}{5}a^{4}-\frac{1}{2}a^{3}-\frac{17}{45}a^{2}-\frac{8}{45}$, $\frac{1}{90}a^{11}-\frac{1}{90}a^{9}-\frac{13}{90}a^{7}+\frac{1}{5}a^{5}-\frac{1}{2}a^{4}-\frac{17}{45}a^{3}-\frac{8}{45}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $7$ | 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: | $\frac{1}{6}a^{11}-\frac{1}{9}a^{10}-\frac{4}{5}a^{9}+\frac{26}{45}a^{8}+\frac{17}{6}a^{7}-\frac{37}{18}a^{6}-\frac{83}{30}a^{5}+\frac{73}{30}a^{4}-\frac{11}{6}a^{3}+\frac{10}{9}a^{2}-\frac{33}{10}a+\frac{56}{45}$, $\frac{7}{18}a^{11}-\frac{7}{90}a^{10}-\frac{173}{90}a^{9}+\frac{37}{90}a^{8}+\frac{125}{18}a^{7}-\frac{67}{45}a^{6}-\frac{227}{30}a^{5}+\frac{53}{30}a^{4}-\frac{43}{18}a^{3}+\frac{73}{90}a^{2}-\frac{563}{90}a+\frac{7}{90}$, $\frac{2}{9}a^{11}+\frac{1}{30}a^{10}-\frac{101}{90}a^{9}-\frac{1}{6}a^{8}+\frac{37}{9}a^{7}+\frac{17}{30}a^{6}-\frac{24}{5}a^{5}-\frac{2}{3}a^{4}-\frac{5}{9}a^{3}-\frac{3}{10}a^{2}-\frac{178}{45}a-\frac{1}{6}$, $\frac{1}{90}a^{11}-\frac{7}{90}a^{10}-\frac{7}{90}a^{9}+\frac{37}{90}a^{8}+\frac{16}{45}a^{7}-\frac{67}{45}a^{6}-\frac{14}{15}a^{5}+\frac{53}{30}a^{4}+\frac{58}{45}a^{3}+\frac{73}{90}a^{2}-\frac{67}{90}a+\frac{26}{45}$, $\frac{1}{6}a^{11}-\frac{1}{9}a^{10}-\frac{4}{5}a^{9}+\frac{26}{45}a^{8}+\frac{17}{6}a^{7}-\frac{37}{18}a^{6}-\frac{83}{30}a^{5}+\frac{73}{30}a^{4}-\frac{11}{6}a^{3}+\frac{10}{9}a^{2}-\frac{23}{10}a+\frac{56}{45}$, $\frac{1}{6}a^{11}+\frac{1}{9}a^{10}-\frac{4}{5}a^{9}-\frac{26}{45}a^{8}+\frac{17}{6}a^{7}+\frac{37}{18}a^{6}-\frac{83}{30}a^{5}-\frac{73}{30}a^{4}-\frac{11}{6}a^{3}-\frac{10}{9}a^{2}-\frac{23}{10}a-\frac{56}{45}$, $\frac{4}{45}a^{11}-\frac{1}{6}a^{10}-\frac{22}{45}a^{9}+\frac{4}{5}a^{8}+\frac{83}{45}a^{7}-\frac{17}{6}a^{6}-\frac{27}{10}a^{5}+\frac{83}{30}a^{4}+\frac{43}{90}a^{3}+\frac{11}{6}a^{2}-\frac{37}{45}a+\frac{23}{10}$ (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: | \( 167.924001727 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 167.924001727 \cdot 1}{2\cdot\sqrt{86161557405625}}\cr\approx \mathstrut & 0.225561755217 \end{aligned}\] (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
\(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\zeta_{13})^+\), 6.2.9282325.1, 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: | 6.2.9282325.1 |
Degree 8 sibling: | 8.0.3016755625.2 |
Degree 12 sibling: | 12.0.2154038935140625.1 |
Minimal sibling: | 6.2.9282325.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |