Normalized defining polynomial
\( x^{12} + 10x^{10} + 25x^{8} - 125x^{4} - 1000x^{2} - 625 \)
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: | \(-2560000000000000000\) \(\medspace = -\,2^{24}\cdot 5^{16}\) | 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: | \(34.20\) | 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: | $2^{247/96}5^{31/20}\approx 72.0988935112507$ | ||
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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{25}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{14125}a^{10}+\frac{26}{2825}a^{8}-\frac{53}{565}a^{6}-\frac{32}{565}a^{4}+\frac{22}{113}a^{2}+\frac{33}{113}$, $\frac{1}{14125}a^{11}+\frac{26}{2825}a^{9}-\frac{39}{2825}a^{7}-\frac{32}{565}a^{5}+\frac{22}{113}a^{3}+\frac{33}{113}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{11}{2825}a^{10}+\frac{74}{2825}a^{8}+\frac{23}{565}a^{6}+\frac{48}{565}a^{4}-\frac{33}{113}a^{2}-\frac{106}{113}$, $\frac{19}{14125}a^{10}+\frac{42}{2825}a^{8}+\frac{2}{113}a^{6}-\frac{43}{565}a^{4}-\frac{34}{113}a^{2}-\frac{51}{113}$, $\frac{2}{2825}a^{10}+\frac{34}{2825}a^{8}+\frac{7}{113}a^{6}+\frac{19}{565}a^{4}-\frac{6}{113}a^{2}-\frac{9}{113}$, $\frac{33}{14125}a^{11}-\frac{93}{14125}a^{10}+\frac{67}{2825}a^{9}-\frac{158}{2825}a^{8}+\frac{182}{2825}a^{7}-\frac{43}{565}a^{6}-\frac{39}{565}a^{5}+\frac{151}{565}a^{4}-\frac{65}{113}a^{3}+\frac{101}{113}a^{2}-\frac{154}{113}a+\frac{321}{113}$, $\frac{66}{14125}a^{11}-\frac{33}{14125}a^{10}+\frac{134}{2825}a^{9}-\frac{67}{2825}a^{8}+\frac{364}{2825}a^{7}-\frac{59}{565}a^{6}+\frac{7}{113}a^{5}-\frac{187}{565}a^{4}-\frac{17}{113}a^{3}-\frac{48}{113}a^{2}-\frac{308}{113}a+\frac{41}{113}$, $\frac{51}{14125}a^{11}+\frac{3}{565}a^{10}+\frac{83}{2825}a^{9}+\frac{142}{2825}a^{8}+\frac{9}{565}a^{7}+\frac{93}{565}a^{6}-\frac{163}{565}a^{5}+\frac{199}{565}a^{4}-\frac{121}{113}a^{3}+\frac{68}{113}a^{2}-\frac{577}{113}a-\frac{124}{113}$ | 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: | \( 19843.2099426 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 19843.2099426 \cdot 2}{2\cdot\sqrt{2560000000000000000}}\cr\approx \mathstrut & 0.485793028141 \end{aligned}\]
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 $C_2\wr S_5$ |
Intermediate fields
6.2.5000000.1 |
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: | 12.2.102400000000000000.11 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }$ | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 | $[4/3, 4/3, 2, 7/3, 7/3, 3]_{3}^{2}$ |
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.10.15.15 | $x^{10} + 10 x^{6} + 10$ | $10$ | $1$ | $15$ | $F_{5}\times C_2$ | $[7/4]_{4}^{2}$ |