Normalized defining polynomial
\( x^{12} + 2x^{10} - 6x^{8} + 30x^{6} - 7x^{4} + 28x^{2} + 25 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(13685690504052736\) \(\medspace = 2^{24}\cdot 13^{8}\) | 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: | \(22.12\) | 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: | $2^{9/4}13^{2/3}\approx 26.299433382699032$ | ||
Ramified primes: | \(2\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{15}a^{9}-\frac{1}{15}a^{7}+\frac{2}{15}a^{5}-\frac{2}{5}a^{3}-\frac{4}{15}a$, $\frac{1}{1185}a^{10}-\frac{2}{395}a^{8}+\frac{14}{395}a^{6}+\frac{484}{1185}a^{4}+\frac{466}{1185}a^{2}-\frac{36}{79}$, $\frac{1}{1185}a^{11}-\frac{2}{395}a^{9}+\frac{14}{395}a^{7}+\frac{484}{1185}a^{5}+\frac{466}{1185}a^{3}-\frac{36}{79}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{14}{395}a^{10}+\frac{143}{1185}a^{8}-\frac{211}{1185}a^{6}+\frac{578}{1185}a^{4}+\frac{1402}{1185}a^{2}-\frac{11}{79}$, $\frac{19}{237}a^{11}+\frac{47}{395}a^{9}-\frac{671}{1185}a^{7}+\frac{1054}{395}a^{5}-\frac{622}{395}a^{3}+\frac{3131}{1185}a$, $\frac{1}{1185}a^{10}-\frac{2}{395}a^{8}+\frac{14}{395}a^{6}+\frac{484}{1185}a^{4}+\frac{466}{1185}a^{2}+\frac{43}{79}$, $\frac{17}{395}a^{11}+\frac{134}{1185}a^{10}+\frac{247}{1185}a^{9}+\frac{127}{395}a^{8}+\frac{3}{395}a^{7}-\frac{692}{1185}a^{6}+\frac{34}{79}a^{5}+\frac{947}{395}a^{4}+\frac{3463}{1185}a^{3}+\frac{933}{395}a^{2}+\frac{2638}{1185}a+\frac{301}{237}$, $\frac{12}{395}a^{11}+\frac{8}{237}a^{10}+\frac{20}{237}a^{9}+\frac{31}{237}a^{8}+\frac{11}{1185}a^{7}+\frac{20}{237}a^{6}+\frac{1466}{1185}a^{5}+\frac{238}{237}a^{4}-\frac{35}{79}a^{3}-\frac{64}{237}a^{2}+\frac{1811}{1185}a+\frac{104}{237}$ | 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: | \( 743.28971555 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 743.28971555 \cdot 1}{2\cdot\sqrt{13685690504052736}}\cr\approx \mathstrut & 0.19546724014 \end{aligned}\]
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
\(\Q(\sqrt{-2}) \), 3.3.169.1, 6.4.14623232.1, 6.0.14623232.1, 6.2.1827904.1 |
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: | 6.4.14623232.1 |
Degree 8 sibling: | 8.0.7487094784.2 |
Degree 12 sibling: | 12.4.13685690504052736.4 |
Minimal sibling: | 6.4.14623232.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.2.0.1}{2} }^{6}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.24.258 | $x^{12} + 10 x^{10} + 4 x^{9} + 142 x^{8} + 184 x^{7} + 720 x^{6} + 1136 x^{5} + 1244 x^{4} + 880 x^{3} - 88 x^{2} - 176 x - 376$ | $4$ | $3$ | $24$ | $A_4 \times C_2$ | $[2, 2, 3]^{3}$ |
\(13\) | 13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |