Normalized defining polynomial
\( x^{12} - 6 x^{11} + 19 x^{10} - 40 x^{9} + 59 x^{8} - 62 x^{7} + 41 x^{6} - 8 x^{5} - 33 x^{4} + \cdots - 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(40777534861441\) \(\medspace = 7^{4}\cdot 19^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(13.62\) | 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))
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Galois root discriminant: | $7^{1/2}19^{2/3}\approx 18.838721275076466$ | ||
Ramified primes: | \(7\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{14}a^{10}+\frac{1}{7}a^{9}-\frac{1}{7}a^{8}+\frac{3}{14}a^{7}+\frac{3}{14}a^{6}+\frac{5}{14}a^{5}+\frac{5}{14}a^{4}-\frac{3}{7}a^{3}-\frac{1}{2}a^{2}+\frac{3}{14}a-\frac{2}{7}$, $\frac{1}{14}a^{11}+\frac{1}{14}a^{9}-\frac{3}{14}a^{7}-\frac{1}{14}a^{6}-\frac{5}{14}a^{5}-\frac{1}{7}a^{4}+\frac{5}{14}a^{3}-\frac{2}{7}a^{2}-\frac{3}{14}a-\frac{3}{7}$
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)
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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)
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Fundamental units: | $\frac{57}{14}a^{11}-\frac{155}{7}a^{10}+\frac{907}{14}a^{9}-\frac{880}{7}a^{8}+\frac{2343}{14}a^{7}-154a^{6}+\frac{521}{7}a^{5}+\frac{205}{14}a^{4}-\frac{898}{7}a^{3}+\frac{2047}{14}a^{2}-\frac{386}{7}a+\frac{65}{14}$, $\frac{8}{7}a^{10}-\frac{40}{7}a^{9}+\frac{110}{7}a^{8}-\frac{200}{7}a^{7}+\frac{489}{14}a^{6}-\frac{403}{14}a^{5}+\frac{129}{14}a^{4}+\frac{107}{14}a^{3}-\frac{67}{2}a^{2}+\frac{391}{14}a-\frac{71}{14}$, $\frac{71}{14}a^{11}-\frac{391}{14}a^{10}+\frac{579}{7}a^{9}-\frac{2277}{14}a^{8}+\frac{441}{2}a^{7}-\frac{2917}{14}a^{6}+\frac{217}{2}a^{5}+\frac{68}{7}a^{4}-\frac{1124}{7}a^{3}+\frac{2719}{14}a^{2}-84a+\frac{79}{7}$, $\frac{71}{7}a^{11}-\frac{781}{14}a^{10}+\frac{2311}{14}a^{9}-\frac{2271}{7}a^{8}+\frac{3078}{7}a^{7}-\frac{831}{2}a^{6}+\frac{3029}{14}a^{5}+\frac{277}{14}a^{4}-\frac{2251}{7}a^{3}+\frac{2712}{7}a^{2}-\frac{2349}{14}a+\frac{333}{14}$, $a-1$, $a$, $\frac{39}{14}a^{11}-\frac{199}{14}a^{10}+\frac{279}{7}a^{9}-\frac{1037}{14}a^{8}+\frac{187}{2}a^{7}-\frac{1133}{14}a^{6}+\frac{65}{2}a^{5}+\frac{90}{7}a^{4}-\frac{555}{7}a^{3}+\frac{1027}{14}a^{2}-21a+\frac{15}{7}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 136.452877594 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 136.452877594 \cdot 1}{2\cdot\sqrt{40777534861441}}\cr\approx \mathstrut & 0.266429110796 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T6):
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
3.3.361.1, 6.4.912247.1 x2, 6.2.6385729.1 |
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: | 6.4.912247.1 |
Degree 8 sibling: | 8.0.312900721.1 |
Degree 12 sibling: | 12.0.1998099208210609.1 |
Minimal sibling: | 6.4.912247.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(19\) | 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |