Normalized defining polynomial
\( x^{12} + 10x^{10} + 55x^{8} + 148x^{6} + 215x^{4} + 118x^{2} + 1 \)
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: | \(672589783760896\) \(\medspace = 2^{18}\cdot 37^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(17.20\) | 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: | $2^{19/12}37^{1/2}\approx 18.227692292212264$ | ||
Ramified primes: | \(2\), \(37\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{5956}a^{10}+\frac{1249}{5956}a^{8}-\frac{497}{2978}a^{6}-\frac{1}{2}a^{5}+\frac{737}{2978}a^{4}-\frac{1991}{5956}a^{2}-\frac{1}{2}a-\frac{947}{5956}$, $\frac{1}{5956}a^{11}-\frac{60}{1489}a^{9}-\frac{1}{4}a^{8}-\frac{497}{2978}a^{7}-\frac{376}{1489}a^{5}-\frac{1}{2}a^{4}+\frac{987}{5956}a^{3}-\frac{1}{2}a^{2}+\frac{271}{2978}a+\frac{1}{4}$
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: | \( -1 \) (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: | $\frac{465}{2978}a^{11}+\frac{2271}{1489}a^{9}+\frac{24693}{2978}a^{7}+\frac{64497}{2978}a^{5}+\frac{45586}{1489}a^{3}+\frac{45059}{2978}a$, $\frac{745}{5956}a^{11}+\frac{18}{1489}a^{10}+\frac{7325}{5956}a^{9}+\frac{147}{1489}a^{8}+\frac{19853}{2978}a^{7}+\frac{1441}{2978}a^{6}+\frac{51739}{2978}a^{5}+\frac{1219}{1489}a^{4}+\frac{142693}{5956}a^{3}+\frac{1285}{2978}a^{2}+\frac{68765}{5956}a-\frac{2156}{1489}$, $\frac{745}{5956}a^{11}-\frac{18}{1489}a^{10}+\frac{7325}{5956}a^{9}-\frac{147}{1489}a^{8}+\frac{19853}{2978}a^{7}-\frac{1441}{2978}a^{6}+\frac{51739}{2978}a^{5}-\frac{1219}{1489}a^{4}+\frac{142693}{5956}a^{3}-\frac{1285}{2978}a^{2}+\frac{68765}{5956}a+\frac{2156}{1489}$, $\frac{905}{2978}a^{11}+\frac{4564}{1489}a^{9}+\frac{25206}{1489}a^{7}+\frac{68407}{1489}a^{5}+\frac{199361}{2978}a^{3}+\frac{54663}{1489}a$, $\frac{140}{1489}a^{11}+\frac{39}{1489}a^{10}+\frac{2783}{2978}a^{9}+\frac{637}{2978}a^{8}+\frac{15013}{2978}a^{7}+\frac{1437}{1489}a^{6}+\frac{38981}{2978}a^{5}+\frac{2393}{1489}a^{4}+\frac{51521}{2978}a^{3}+\frac{1268}{1489}a^{2}+\frac{11853}{1489}a+\frac{2073}{2978}$ | 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: | \( 176.43713101 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 176.43713101 \cdot 2}{2\cdot\sqrt{672589783760896}}\cr\approx \mathstrut & 0.41859502670 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T24):
A solvable group of order 48 |
The 10 conjugacy class representatives for $C_2 \times S_4$ |
Character table for $C_2 \times S_4$ |
Intermediate fields
\(\Q(\sqrt{-37}) \), 3.3.148.1, 6.4.350464.1, 6.0.12967168.1, 6.2.3241792.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.4.350464.1, 6.4.12967168.3 |
Degree 8 siblings: | 8.0.5607424.1, 8.0.7676563456.4 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.4.350464.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.79 | $x^{12} + 2 x^{9} + 2 x^{7} + 2 x^{2} + 6$ | $12$ | $1$ | $18$ | $C_2 \times S_4$ | $[4/3, 4/3, 2]_{3}^{2}$ |
\(37\) | 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |