Normalized defining polynomial
\( x^{12} - 2x^{11} + 3x^{10} - x^{9} + 2x^{6} - x^{3} + 3x^{2} - 2x + 1 \)
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: | \(181824635281\) \(\medspace = 653^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(8.68\) | 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: | $653^{1/2}\approx 25.553864678361276$ | ||
Ramified primes: | \(653\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{8}a^{11}+\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{8}a^{8}+\frac{3}{8}a^{7}+\frac{1}{8}a^{6}-\frac{3}{8}a^{5}-\frac{1}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{3}{8}a-\frac{3}{8}$
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)
| |
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)
| |
Fundamental units: | $a$, $\frac{17}{8}a^{11}-\frac{23}{8}a^{10}+\frac{15}{4}a^{9}+\frac{9}{8}a^{8}-\frac{5}{8}a^{7}-\frac{7}{8}a^{6}+\frac{29}{8}a^{5}+\frac{15}{8}a^{4}+\frac{5}{8}a^{3}-\frac{13}{4}a^{2}+\frac{29}{8}a-\frac{11}{8}$, $\frac{5}{4}a^{11}-\frac{7}{4}a^{10}+\frac{5}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{3}{4}a^{6}+\frac{9}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{3}{2}a^{2}+\frac{9}{4}a-\frac{7}{4}$, $\frac{1}{4}a^{11}-\frac{7}{4}a^{10}+\frac{3}{2}a^{9}-\frac{7}{4}a^{8}-\frac{9}{4}a^{7}-\frac{3}{4}a^{6}+\frac{1}{4}a^{5}-\frac{9}{4}a^{4}-\frac{11}{4}a^{3}-\frac{3}{2}a^{2}+\frac{5}{4}a-\frac{7}{4}$, $\frac{3}{8}a^{11}-\frac{5}{8}a^{10}+\frac{1}{4}a^{9}+\frac{3}{8}a^{8}-\frac{7}{8}a^{7}-\frac{5}{8}a^{6}-\frac{1}{8}a^{5}+\frac{5}{8}a^{4}-\frac{1}{8}a^{3}-\frac{7}{4}a^{2}-\frac{1}{8}a+\frac{7}{8}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 3.37086598412 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3.37086598412 \cdot 1}{2\cdot\sqrt{181824635281}}\cr\approx \mathstrut & 0.243200430019 \end{aligned}\]
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
6.2.426409.1 |
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: | 10.2.118731486838493.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(653\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |