Normalized defining polynomial
\( x^{12} - 7x^{10} - 5x^{9} + 20x^{8} + 21x^{7} - 16x^{6} - 22x^{5} + 10x^{4} + 11x^{3} + 7x^{2} + 2x + 1 \)
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: | \(1636073786281\) \(\medspace = 11^{6}\cdot 31^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.42\) | 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: | $11^{1/2}31^{1/2}\approx 18.466185312619388$ | ||
Ramified primes: | \(11\), \(31\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}-\frac{2}{5}$, $\frac{1}{5}a^{10}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a$, $\frac{1}{8235}a^{11}+\frac{677}{8235}a^{10}+\frac{152}{2745}a^{9}-\frac{929}{8235}a^{8}+\frac{1888}{8235}a^{7}+\frac{25}{1647}a^{6}+\frac{68}{915}a^{5}+\frac{2552}{8235}a^{4}+\frac{11}{8235}a^{3}+\frac{58}{549}a^{2}-\frac{629}{8235}a+\frac{4036}{8235}$
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)
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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: | $\frac{1168}{8235}a^{11}+\frac{176}{8235}a^{10}-\frac{617}{549}a^{9}-\frac{6287}{8235}a^{8}+\frac{29497}{8235}a^{7}+\frac{27416}{8235}a^{6}-\frac{4024}{915}a^{5}-\frac{31627}{8235}a^{4}+\frac{32612}{8235}a^{3}+\frac{4379}{2745}a^{2}-\frac{9992}{8235}a+\frac{334}{8235}$, $\frac{212}{1647}a^{11}-\frac{472}{8235}a^{10}-\frac{2482}{2745}a^{9}-\frac{3128}{8235}a^{8}+\frac{4976}{1647}a^{7}+\frac{20504}{8235}a^{6}-\frac{3499}{915}a^{5}-\frac{7427}{1647}a^{4}+\frac{24836}{8235}a^{3}+\frac{9293}{2745}a^{2}-\frac{4646}{8235}a-\frac{2393}{8235}$, $\frac{449}{8235}a^{11}-\frac{722}{8235}a^{10}-\frac{926}{2745}a^{9}+\frac{2864}{8235}a^{8}+\frac{9389}{8235}a^{7}-\frac{6461}{8235}a^{6}-\frac{262}{183}a^{5}+\frac{7771}{8235}a^{4}+\frac{1976}{1647}a^{3}-\frac{2099}{2745}a^{2}-\frac{2431}{8235}a-\frac{4477}{8235}$, $\frac{274}{1647}a^{11}-\frac{1418}{8235}a^{10}-\frac{2576}{2745}a^{9}+\frac{2048}{8235}a^{8}+\frac{20534}{8235}a^{7}+\frac{1609}{8235}a^{6}-\frac{217}{183}a^{5}+\frac{1306}{8235}a^{4}-\frac{4694}{8235}a^{3}-\frac{176}{2745}a^{2}+\frac{16121}{8235}a+\frac{341}{8235}$, $\frac{659}{2745}a^{11}-\frac{743}{2745}a^{10}-\frac{1397}{915}a^{9}+\frac{424}{549}a^{8}+\frac{13334}{2745}a^{7}-\frac{3269}{2745}a^{6}-\frac{1792}{305}a^{5}+\frac{5671}{2745}a^{4}+\frac{13288}{2745}a^{3}-\frac{391}{183}a^{2}+\frac{4376}{2745}a-\frac{181}{2745}$ | 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: | \( 10.2890687023 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 10.2890687023 \cdot 1}{2\cdot\sqrt{1636073786281}}\cr\approx \mathstrut & 0.247470729444 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T23):
A solvable group of order 48 |
The 10 conjugacy class representatives for $C_2 \times S_4$ |
Character table for $C_2 \times S_4$ |
Intermediate fields
\(\Q(\sqrt{-11}) \), 3.1.31.1, 6.0.1279091.1, 6.0.10571.1, 6.2.116281.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.327701.1, 6.0.10571.1 |
Degree 8 siblings: | 8.4.13521270961.1, 8.0.14070001.1 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.10571.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |