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: | $[4, 4]$ | 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: | $7$ | 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: | $\frac{49}{1185}a^{10}-\frac{101}{1185}a^{8}-\frac{104}{395}a^{6}-\frac{1201}{1185}a^{4}-\frac{157}{395}a^{2}-\frac{238}{237}$, $\frac{1}{1185}a^{10}+\frac{2}{395}a^{8}+\frac{14}{395}a^{6}-\frac{484}{1185}a^{4}+\frac{466}{1185}a^{2}-\frac{43}{79}$, $\frac{19}{1185}a^{11}-\frac{44}{1185}a^{9}-\frac{10}{79}a^{7}-\frac{427}{1185}a^{5}+\frac{2}{395}a^{3}+\frac{622}{1185}a$, $\frac{26}{1185}a^{11}-\frac{2}{1185}a^{9}-\frac{251}{1185}a^{7}-\frac{70}{79}a^{5}-\frac{1867}{1185}a^{3}-\frac{733}{1185}a+1$, $\frac{83}{1185}a^{11}+\frac{92}{1185}a^{10}-\frac{134}{1185}a^{9}-\frac{238}{1185}a^{8}-\frac{701}{1185}a^{7}-\frac{481}{1185}a^{6}-\frac{777}{395}a^{5}-\frac{2263}{1185}a^{4}-\frac{664}{1185}a^{3}+\frac{1397}{1185}a^{2}-\frac{52}{1185}a-\frac{334}{237}$, $\frac{1}{237}a^{11}-\frac{16}{395}a^{10}-\frac{49}{1185}a^{9}+\frac{107}{1185}a^{8}+\frac{131}{1185}a^{7}+\frac{118}{395}a^{6}-\frac{208}{1185}a^{5}+\frac{239}{395}a^{4}+\frac{671}{1185}a^{3}+\frac{937}{1185}a^{2}+\frac{646}{1185}a+\frac{109}{237}$, $\frac{35}{237}a^{11}+\frac{107}{1185}a^{10}-\frac{293}{1185}a^{9}-\frac{148}{1185}a^{8}-\frac{1103}{1185}a^{7}-\frac{641}{1185}a^{6}-\frac{5621}{1185}a^{5}-\frac{1331}{395}a^{4}-\frac{3533}{1185}a^{3}-\frac{3068}{1185}a^{2}-\frac{6778}{1185}a-\frac{335}{79}$ | 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: | \( 2659.85170055 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 2659.85170055 \cdot 1}{2\cdot\sqrt{13685690504052736}}\cr\approx \mathstrut & 0.283487247072 \end{aligned}\]
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, 6.2.14623232.3, 6.6.14623232.1, 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: | 6.2.14623232.3 |
Degree 8 sibling: | 8.0.7487094784.3 |
Degree 12 sibling: | 12.0.13685690504052736.9 |
Minimal sibling: | 6.2.14623232.3 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{6}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.1.0.1}{1} }^{12}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.24.123 | $x^{12} + 6 x^{10} + 4 x^{9} + 50 x^{8} + 136 x^{7} + 224 x^{6} + 288 x^{5} + 140 x^{4} + 592 x^{3} + 664 x^{2} + 1776 x + 632$ | $4$ | $3$ | $24$ | $A_4 \times C_2$ | $[2, 2, 3]^{3}$ |
\(13\) | 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 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |