Normalized defining polynomial
\( x^{12} - 6 x^{11} + 23 x^{10} - 60 x^{9} + 95 x^{8} - 86 x^{7} - 63 x^{6} + 304 x^{5} - 467 x^{4} + 414 x^{3} - 75 x^{2} - 80 x + 46 \)
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: | \(223580268118933504\) \(\medspace = 2^{18}\cdot 31^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(27.91\) | 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: | $2^{3/2}31^{2/3}\approx 27.911689339648817$ | ||
Ramified primes: | \(2\), \(31\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{6}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{388}a^{10}-\frac{5}{388}a^{9}-\frac{7}{194}a^{8}+\frac{43}{194}a^{7}-\frac{25}{194}a^{6}-\frac{43}{97}a^{5}-\frac{187}{388}a^{4}-\frac{5}{388}a^{3}+\frac{20}{97}a^{2}-\frac{61}{194}a+\frac{14}{97}$, $\frac{1}{105148}a^{11}+\frac{65}{52574}a^{10}+\frac{4651}{52574}a^{9}-\frac{7333}{105148}a^{8}-\frac{3027}{26287}a^{7}+\frac{12286}{26287}a^{6}-\frac{46493}{105148}a^{5}+\frac{5805}{52574}a^{4}-\frac{1240}{26287}a^{3}+\frac{51321}{105148}a^{2}+\frac{6227}{26287}a-\frac{1943}{52574}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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)
| |
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: | $\frac{7}{194}a^{10}-\frac{35}{194}a^{9}+\frac{48}{97}a^{8}-\frac{87}{97}a^{7}+\frac{19}{97}a^{6}+\frac{174}{97}a^{5}-\frac{727}{194}a^{4}+\frac{741}{194}a^{3}+\frac{183}{97}a^{2}-\frac{330}{97}a+\frac{99}{97}$, $\frac{3133}{105148}a^{11}-\frac{3461}{26287}a^{10}+\frac{46167}{105148}a^{9}-\frac{97011}{105148}a^{8}+\frac{20650}{26287}a^{7}+\frac{3003}{52574}a^{6}-\frac{359957}{105148}a^{5}+\frac{128717}{26287}a^{4}-\frac{369371}{105148}a^{3}+\frac{26615}{105148}a^{2}+\frac{330693}{52574}a-\frac{4011}{52574}$, $\frac{1575}{105148}a^{11}-\frac{9819}{52574}a^{10}+\frac{79251}{105148}a^{9}-\frac{206499}{105148}a^{8}+\frac{81749}{26287}a^{7}-\frac{35569}{52574}a^{6}-\frac{563787}{105148}a^{5}+\frac{681713}{52574}a^{4}-\frac{1248651}{105148}a^{3}-\frac{209265}{105148}a^{2}+\frac{312533}{52574}a-\frac{142639}{52574}$, $\frac{6737}{105148}a^{11}-\frac{27975}{105148}a^{10}+\frac{18947}{26287}a^{9}-\frac{32910}{26287}a^{8}+\frac{1057}{52574}a^{7}+\frac{52639}{26287}a^{6}-\frac{523487}{105148}a^{5}+\frac{442009}{105148}a^{4}+\frac{154131}{52574}a^{3}-\frac{30508}{26287}a^{2}+\frac{770}{26287}a+\frac{4668}{26287}$, $\frac{6737}{105148}a^{11}-\frac{11533}{26287}a^{10}+\frac{166573}{105148}a^{9}-\frac{201591}{52574}a^{8}+\frac{135893}{26287}a^{7}-\frac{71685}{52574}a^{6}-\frac{975515}{105148}a^{5}+\frac{565037}{26287}a^{4}-\frac{2124505}{105148}a^{3}+\frac{105805}{26287}a^{2}+\frac{267933}{52574}a-\frac{65521}{26287}$, $\frac{285}{52574}a^{11}-\frac{3135}{105148}a^{10}+\frac{10323}{105148}a^{9}-\frac{22941}{105148}a^{8}+\frac{10161}{52574}a^{7}+\frac{6993}{52574}a^{6}-\frac{31416}{26287}a^{5}+\frac{241341}{105148}a^{4}-\frac{338065}{105148}a^{3}+\frac{283341}{105148}a^{2}-\frac{71609}{52574}a+\frac{15785}{52574}$, $\frac{2411}{105148}a^{11}-\frac{1549}{52574}a^{10}+\frac{9731}{105148}a^{9}+\frac{177}{105148}a^{8}-\frac{13617}{26287}a^{7}-\frac{6993}{52574}a^{6}-\frac{135851}{105148}a^{5}-\frac{97771}{52574}a^{4}-\frac{97703}{105148}a^{3}-\frac{5837}{105148}a^{2}+\frac{99251}{52574}a-\frac{35839}{52574}$ | 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: | \( 52798.990549 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 52798.990549 \cdot 1}{2\cdot\sqrt{223580268118933504}}\cr\approx \mathstrut & 1.3922535808 \end{aligned}\]
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
\(\Q(\sqrt{2}) \), 3.3.961.1, 6.2.7388168.1, 6.6.472842752.1, 6.2.59105344.3 |
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: | 6.2.7388168.1 |
Degree 8 sibling: | 8.0.3782742016.5 |
Degree 12 sibling: | 12.0.3493441689358336.8 |
Minimal sibling: | 6.2.7388168.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
\(31\) | 31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |