Normalized defining polynomial
\( x^{12} + 6x^{10} - 10x^{8} - 60x^{7} - 130x^{6} - 60x^{5} - 10x^{4} + 6x^{2} + 1 \)
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: | \(40000000000000000\) \(\medspace = 2^{18}\cdot 5^{16}\) | 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: | \(24.18\) | 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: | $2^{2}5^{8/5}\approx 52.53055608807534$ | ||
Ramified primes: | \(2\), \(5\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{8}a^{2}-\frac{3}{8}a-\frac{1}{8}$, $\frac{1}{8}a^{9}+\frac{1}{8}a^{6}-\frac{1}{2}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a+\frac{1}{8}$, $\frac{1}{160}a^{10}-\frac{1}{16}a^{9}+\frac{1}{32}a^{8}-\frac{1}{8}a^{7}+\frac{1}{32}a^{6}+\frac{3}{16}a^{5}+\frac{1}{32}a^{4}-\frac{3}{8}a^{3}+\frac{9}{32}a^{2}+\frac{3}{16}a-\frac{79}{160}$, $\frac{1}{1280}a^{11}-\frac{3}{1280}a^{10}+\frac{3}{256}a^{9}-\frac{9}{256}a^{8}+\frac{25}{256}a^{7}+\frac{41}{256}a^{6}-\frac{21}{256}a^{5}+\frac{51}{256}a^{4}+\frac{101}{256}a^{3}-\frac{47}{256}a^{2}-\frac{569}{1280}a-\frac{213}{1280}$
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: | $a$, $\frac{61}{128}a^{11}+\frac{9}{640}a^{10}+\frac{363}{128}a^{9}+\frac{19}{128}a^{8}-\frac{631}{128}a^{7}-\frac{3627}{128}a^{6}-\frac{8029}{128}a^{5}-\frac{3777}{128}a^{4}-\frac{811}{128}a^{3}-\frac{691}{128}a^{2}+\frac{131}{128}a-\frac{41}{640}$, $\frac{397}{1280}a^{11}-\frac{343}{1280}a^{10}+\frac{455}{256}a^{9}-\frac{357}{256}a^{8}-\frac{955}{256}a^{7}-\frac{3739}{256}a^{6}-\frac{6193}{256}a^{5}+\frac{4967}{256}a^{4}+\frac{3105}{256}a^{3}-\frac{3283}{256}a^{2}+\frac{4667}{1280}a-\frac{1153}{1280}$, $\frac{331}{1280}a^{11}-\frac{329}{1280}a^{10}+\frac{433}{256}a^{9}-\frac{427}{256}a^{8}-\frac{429}{256}a^{7}-\frac{3525}{256}a^{6}-\frac{4887}{256}a^{5}+\frac{2633}{256}a^{4}+\frac{503}{256}a^{3}+\frac{1715}{256}a^{2}+\frac{1661}{1280}a+\frac{1041}{1280}$, $\frac{3}{32}a^{11}+\frac{1}{32}a^{10}+\frac{21}{32}a^{9}+\frac{7}{32}a^{8}-\frac{13}{32}a^{7}-\frac{183}{32}a^{6}-\frac{487}{32}a^{5}-\frac{497}{32}a^{4}-\frac{537}{32}a^{3}-\frac{291}{32}a^{2}-\frac{35}{32}a-\frac{37}{32}$, $\frac{65}{64}a^{11}+\frac{1}{64}a^{10}+\frac{391}{64}a^{9}+\frac{7}{64}a^{8}-\frac{643}{64}a^{7}-\frac{3903}{64}a^{6}-\frac{8513}{64}a^{5}-\frac{4093}{64}a^{4}-\frac{903}{64}a^{3}-\frac{263}{64}a^{2}+\frac{127}{64}a-\frac{193}{64}$, $\frac{641}{640}a^{11}-\frac{383}{640}a^{10}+\frac{763}{128}a^{9}-\frac{485}{128}a^{8}-\frac{1303}{128}a^{7}-\frac{7091}{128}a^{6}-\frac{11901}{128}a^{5}+\frac{2807}{128}a^{4}+\frac{5557}{128}a^{3}+\frac{3541}{128}a^{2}+\frac{5151}{640}a-\frac{433}{640}$ | 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: | \( 22656.4836805 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 22656.4836805 \cdot 1}{2\cdot\sqrt{40000000000000000}}\cr\approx \mathstrut & 1.41244638806 \end{aligned}\]
Galois group
$C_2\times A_5$ (as 12T76):
A non-solvable group of order 120 |
The 10 conjugacy class representatives for $C_2\times A_5$ |
Character table for $C_2\times A_5$ |
Intermediate fields
6.2.100000000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 10.0.80000000000000000.12 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.8.3 | $x^{4} + 6 x^{2} + 4 x + 14$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
5.5.8.6 | $x^{5} + 5 x^{4} + 5$ | $5$ | $1$ | $8$ | $D_{5}$ | $[2]^{2}$ | |
5.5.8.6 | $x^{5} + 5 x^{4} + 5$ | $5$ | $1$ | $8$ | $D_{5}$ | $[2]^{2}$ |