Normalized defining polynomial
\( x^{12} - 4x^{11} + 25x^{9} - 65x^{8} + 101x^{7} - 105x^{6} + 68x^{5} - 43x^{4} + 33x^{3} - 10x^{2} + 8x - 8 \)
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: | \(120028742912169\) \(\medspace = 3^{6}\cdot 7^{8}\cdot 13^{4}\) | 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: | \(14.90\) | 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: | $3^{1/2}7^{2/3}13^{1/2}\approx 22.852356834621286$ | ||
Ramified primes: | \(3\), \(7\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{51412}a^{11}+\frac{2250}{12853}a^{10}+\frac{2672}{12853}a^{9}-\frac{8487}{51412}a^{8}+\frac{6925}{51412}a^{7}+\frac{15751}{51412}a^{6}+\frac{1897}{51412}a^{5}-\frac{6917}{25706}a^{4}-\frac{15809}{51412}a^{3}-\frac{10081}{51412}a^{2}-\frac{362}{12853}a+\frac{5216}{12853}$
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: | $\frac{473}{51412}a^{11}-\frac{2549}{12853}a^{10}+\frac{4262}{12853}a^{9}+\frac{47197}{51412}a^{8}-\frac{169079}{51412}a^{7}+\frac{303955}{51412}a^{6}-\frac{388019}{51412}a^{5}+\frac{121451}{25706}a^{4}-\frac{125741}{51412}a^{3}+\frac{115827}{51412}a^{2}+\frac{8716}{12853}a-\frac{608}{12853}$, $\frac{1875}{25706}a^{11}-\frac{989}{25706}a^{10}-\frac{5340}{12853}a^{9}+\frac{5871}{12853}a^{8}-\frac{22861}{25706}a^{7}+\frac{48343}{25706}a^{6}-\frac{41965}{25706}a^{5}+\frac{37861}{12853}a^{4}-\frac{20708}{12853}a^{3}-\frac{33671}{25706}a^{2}-\frac{28723}{25706}a-\frac{2266}{12853}$, $\frac{15739}{51412}a^{11}-\frac{10118}{12853}a^{10}-\frac{13261}{12853}a^{9}+\frac{299955}{51412}a^{8}-\frac{592103}{51412}a^{7}+\frac{844623}{51412}a^{6}-\frac{758963}{51412}a^{5}+\frac{134080}{12853}a^{4}-\frac{523597}{51412}a^{3}+\frac{223927}{51412}a^{2}-\frac{20131}{25706}a+\frac{28219}{12853}$, $\frac{31853}{51412}a^{11}-\frac{24784}{12853}a^{10}-\frac{41259}{25706}a^{9}+\frac{707653}{51412}a^{8}-\frac{1466403}{51412}a^{7}+\frac{2043375}{51412}a^{6}-\frac{1731967}{51412}a^{5}+\frac{410221}{25706}a^{4}-\frac{651893}{51412}a^{3}+\frac{292025}{51412}a^{2}+\frac{11208}{12853}a+\frac{58782}{12853}$, $\frac{561}{12853}a^{11}-\frac{2229}{12853}a^{10}+\frac{87}{25706}a^{9}+\frac{27365}{25706}a^{8}-\frac{35240}{12853}a^{7}+\frac{57712}{12853}a^{6}-\frac{66847}{12853}a^{5}+\frac{53750}{12853}a^{4}-\frac{51691}{12853}a^{3}+\frac{64023}{25706}a^{2}-\frac{18031}{25706}a+\frac{8474}{12853}$, $\frac{1212}{12853}a^{11}-\frac{4197}{12853}a^{10}-\frac{1968}{12853}a^{9}+\frac{56577}{25706}a^{8}-\frac{64174}{12853}a^{7}+\frac{93478}{12853}a^{6}-\frac{91494}{12853}a^{5}+\frac{57769}{12853}a^{4}-\frac{48097}{12853}a^{3}+\frac{43590}{12853}a^{2}-\frac{26789}{25706}a+\frac{5317}{12853}$, $\frac{473}{51412}a^{11}-\frac{2549}{12853}a^{10}+\frac{4262}{12853}a^{9}+\frac{47197}{51412}a^{8}-\frac{169079}{51412}a^{7}+\frac{303955}{51412}a^{6}-\frac{388019}{51412}a^{5}+\frac{121451}{25706}a^{4}-\frac{125741}{51412}a^{3}+\frac{115827}{51412}a^{2}-\frac{4137}{12853}a+\frac{12245}{12853}$ | 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: | \( 268.508625726 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 268.508625726 \cdot 1}{2\cdot\sqrt{120028742912169}}\cr\approx \mathstrut & 0.305580103307 \end{aligned}\]
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
\(\Q(\zeta_{7})^+\), 6.4.842751.1 x2, 6.2.405769.1 |
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: | 6.4.842751.1 |
Degree 8 sibling: | 8.0.5554571841.5 |
Degree 12 sibling: | 12.0.20284857552156561.2 |
Minimal sibling: | 6.4.842751.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |