Normalized defining polynomial
\( x^{12} - 10x^{10} + 32x^{8} + 32x^{6} - 272x^{4} - 128x^{2} - 16 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-7058296125839865020416\) \(\medspace = -\,2^{24}\cdot 29^{10}\) | 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: | \(66.18\) | 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: | $2^{247/96}29^{5/6}\approx 98.44419029753239$ | ||
Ramified primes: | \(2\), \(29\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{4}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{8}a^{9}$, $\frac{1}{248}a^{10}+\frac{2}{31}a^{8}+\frac{7}{124}a^{6}+\frac{3}{31}a^{4}+\frac{13}{31}a^{2}+\frac{12}{31}$, $\frac{1}{248}a^{11}-\frac{15}{248}a^{9}+\frac{7}{124}a^{7}+\frac{3}{31}a^{5}-\frac{5}{62}a^{3}+\frac{12}{31}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $6$ | 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: | $\frac{5}{62}a^{11}-\frac{207}{248}a^{9}+\frac{357}{124}a^{7}+\frac{89}{62}a^{5}-\frac{670}{31}a^{3}-\frac{132}{31}a$, $\frac{4}{31}a^{11}-\frac{325}{248}a^{9}+\frac{267}{62}a^{7}+\frac{223}{62}a^{5}-\frac{1103}{31}a^{3}-\frac{360}{31}a$, $\frac{1}{2}a^{3}$, $\frac{357}{124}a^{11}-\frac{110}{31}a^{10}-\frac{897}{31}a^{9}+\frac{1123}{31}a^{8}+\frac{2908}{31}a^{7}-\frac{3772}{31}a^{6}+\frac{2669}{31}a^{5}-\frac{2578}{31}a^{4}-\frac{24508}{31}a^{3}+\frac{30410}{31}a^{2}-\frac{9722}{31}a+\frac{6521}{31}$, $\frac{473}{124}a^{11}-\frac{51}{248}a^{10}-\frac{2447}{62}a^{9}+\frac{199}{62}a^{8}+\frac{8271}{62}a^{7}-\frac{1721}{124}a^{6}+\frac{5521}{62}a^{5}+\frac{250}{31}a^{4}-\frac{33737}{31}a^{3}+\frac{3770}{31}a^{2}-\frac{7992}{31}a-\frac{240}{31}$, $\frac{21835}{248}a^{11}+\frac{443}{124}a^{10}-\frac{111915}{124}a^{9}-\frac{1080}{31}a^{8}+\frac{94267}{31}a^{7}+\frac{3085}{31}a^{6}+\frac{64327}{31}a^{5}+\frac{11423}{62}a^{4}-\frac{759698}{31}a^{3}-\frac{31727}{31}a^{2}-\frac{164354}{31}a-\frac{33109}{31}$ (assuming GRH) | 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: | \( 1097642.20502 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 1097642.20502 \cdot 2}{2\cdot\sqrt{7058296125839865020416}}\cr\approx \mathstrut & 0.511764543254 \end{aligned}\] (assuming GRH)
Galois group
$C_2\wr S_5$ (as 12T270):
A non-solvable group of order 7680 |
The 37 conjugacy class representatives for $C_2\wr S_5$ |
Character table for $C_2\wr S_5$ is not computed |
Intermediate fields
6.2.1312713536.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 12.2.7058296125839865020416.47 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.24.388 | $x^{12} + 4 x^{11} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 6 x^{2} + 4 x + 14$ | $12$ | $1$ | $24$ | 12T149 | $[4/3, 4/3, 2, 7/3, 7/3, 3]_{3}^{2}$ |
\(29\) | 29.12.10.1 | $x^{12} + 144 x^{11} + 8652 x^{10} + 277920 x^{9} + 5045820 x^{8} + 49440384 x^{7} + 211217114 x^{6} + 98884944 x^{5} + 20432100 x^{4} + 10157760 x^{3} + 142459992 x^{2} + 1361530944 x + 5427130041$ | $6$ | $2$ | $10$ | $D_6$ | $[\ ]_{6}^{2}$ |