Normalized defining polynomial
\( x^{12} - 4 x^{11} - 6 x^{10} + 38 x^{9} + 2 x^{8} - 134 x^{7} + 49 x^{6} + 213 x^{5} - 104 x^{4} + \cdots - 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(3185030335661329\) \(\medspace = 17^{2}\cdot 101^{2}\cdot 32869^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.59\) | 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))
| |
Galois root discriminant: | $17^{1/2}101^{1/2}32869^{1/2}\approx 7512.394624884931$ | ||
Ramified primes: | \(17\), \(101\), \(32869\) | 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)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $11$ | 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: | $a^{11}-3a^{10}-8a^{9}+26a^{8}+25a^{7}-82a^{6}-43a^{5}+112a^{4}+46a^{3}-57a^{2}-22a+1$, $a^{11}-3a^{10}-8a^{9}+26a^{8}+25a^{7}-83a^{6}-41a^{5}+116a^{4}+39a^{3}-60a^{2}-18a+1$, $a^{4}-a^{3}-4a^{2}+2a+2$, $a^{9}-2a^{8}-8a^{7}+14a^{6}+23a^{5}-31a^{4}-28a^{3}+22a^{2}+13a-1$, $a^{11}-2a^{10}-11a^{9}+20a^{8}+45a^{7}-70a^{6}-83a^{5}+100a^{4}+65a^{3}-49a^{2}-17a+1$, $a^{6}-2a^{5}-5a^{4}+9a^{3}+7a^{2}-9a-3$, $a^{2}-2$, $a^{11}-3a^{10}-9a^{9}+28a^{8}+32a^{7}-95a^{6}-58a^{5}+137a^{4}+55a^{3}-70a^{2}-23a+1$, $a^{11}-3a^{10}-9a^{9}+29a^{8}+30a^{7}-101a^{6}-47a^{5}+148a^{4}+37a^{3}-75a^{2}-14a$, $a^{11}-3a^{10}-8a^{9}+27a^{8}+22a^{7}-87a^{6}-24a^{5}+117a^{4}+10a^{3}-54a^{2}-4a$, $a^{10}-4a^{9}-4a^{8}+30a^{7}-5a^{6}-76a^{5}+32a^{4}+74a^{3}-24a^{2}-24a-1$ | 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: | \( 5379.62836167 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 5379.62836167 \cdot 1}{2\cdot\sqrt{3185030335661329}}\cr\approx \mathstrut & 0.195220508782 \end{aligned}\]
Galois group
$C_2^5:S_6$ (as 12T285):
A non-solvable group of order 23040 |
The 37 conjugacy class representatives for $C_2^5:S_6$ |
Character table for $C_2^5:S_6$ |
Intermediate fields
6.6.3319769.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 32 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | 17.4.2.2 | $x^{4} - 272 x^{2} + 867$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
17.8.0.1 | $x^{8} + 11 x^{4} + 12 x^{3} + 6 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
\(101\) | 101.2.1.2 | $x^{2} + 202$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
101.2.1.2 | $x^{2} + 202$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
101.2.0.1 | $x^{2} + 97 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
101.6.0.1 | $x^{6} + 2 x^{4} + 90 x^{3} + 20 x^{2} + 67 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(32869\) | $\Q_{32869}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{32869}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |