Normalized defining polynomial
\( x^{12} - 13x^{6} + 49 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(47038089498624\) \(\medspace = 2^{12}\cdot 3^{14}\cdot 7^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.78\) | 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: | $2\cdot 3^{7/6}7^{2/3}\approx 26.36757274471889$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{2}$, $\frac{1}{21}a^{9}+\frac{1}{21}a^{3}$, $\frac{1}{63}a^{10}-\frac{1}{9}a^{8}+\frac{1}{9}a^{6}-\frac{20}{63}a^{4}+\frac{2}{9}a^{2}-\frac{2}{9}$, $\frac{1}{63}a^{11}-\frac{1}{63}a^{9}+\frac{1}{9}a^{7}-\frac{20}{63}a^{5}+\frac{20}{63}a^{3}-\frac{2}{9}a$
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: | \( \frac{2}{21} a^{9} - \frac{19}{21} a^{3} \) (order $12$) | 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: | $\frac{1}{7}a^{9}-\frac{1}{3}a^{6}-\frac{6}{7}a^{3}+\frac{8}{3}$, $\frac{1}{63}a^{10}-\frac{1}{9}a^{8}-\frac{2}{9}a^{6}-\frac{20}{63}a^{4}+\frac{2}{9}a^{2}+\frac{13}{9}$, $\frac{5}{63}a^{10}+\frac{1}{9}a^{8}-\frac{1}{9}a^{6}-\frac{37}{63}a^{4}-\frac{2}{9}a^{2}+\frac{11}{9}$, $\frac{5}{63}a^{10}-\frac{1}{21}a^{9}-\frac{2}{9}a^{8}-\frac{1}{9}a^{6}-\frac{37}{63}a^{4}-\frac{1}{21}a^{3}+\frac{13}{9}a^{2}+\frac{2}{9}$, $\frac{4}{63}a^{10}-\frac{1}{7}a^{9}+\frac{2}{9}a^{8}-\frac{2}{9}a^{6}-\frac{17}{63}a^{4}+\frac{6}{7}a^{3}-\frac{13}{9}a^{2}+\frac{13}{9}$ | 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: | \( 272.5550622157524 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 272.5550622157524 \cdot 1}{12\cdot\sqrt{47038089498624}}\cr\approx \mathstrut & 0.203763963517001 \end{aligned}\]
Galois group
$C_6\times S_3$ (as 12T18):
A solvable group of order 36 |
The 18 conjugacy class representatives for $C_6\times S_3$ |
Character table for $C_6\times S_3$ |
Intermediate fields
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 6.0.107163.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 36 |
Degree 18 siblings: | 18.6.37954454265953846507077632.1, 18.0.1405720528368660981743616.1 |
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 | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
\(3\) | 3.12.14.15 | $x^{12} - 6 x^{9} + 6 x^{8} + 24 x^{6} - 18 x^{5} + 9 x^{4} - 18 x^{3} + 18 x^{2} + 9$ | $6$ | $2$ | $14$ | $C_6\times S_3$ | $[3/2]_{2}^{6}$ |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.21.6t1.a.a | $1$ | $ 3 \cdot 7 $ | 6.0.64827.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.84.6t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 7 $ | 6.6.4148928.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.28.6t1.a.a | $1$ | $ 2^{2} \cdot 7 $ | 6.0.153664.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.21.6t1.a.b | $1$ | $ 3 \cdot 7 $ | 6.0.64827.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.28.6t1.a.b | $1$ | $ 2^{2} \cdot 7 $ | 6.0.153664.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.84.6t1.a.b | $1$ | $ 2^{2} \cdot 3 \cdot 7 $ | 6.6.4148928.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
2.1323.3t2.b.a | $2$ | $ 3^{3} \cdot 7^{2}$ | 3.1.1323.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.21168.6t3.h.a | $2$ | $ 2^{4} \cdot 3^{3} \cdot 7^{2}$ | 6.0.112021056.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.189.6t5.a.a | $2$ | $ 3^{3} \cdot 7 $ | 6.0.107163.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.3024.12t18.b.a | $2$ | $ 2^{4} \cdot 3^{3} \cdot 7 $ | 12.0.47038089498624.3 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.189.6t5.a.b | $2$ | $ 3^{3} \cdot 7 $ | 6.0.107163.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.3024.12t18.b.b | $2$ | $ 2^{4} \cdot 3^{3} \cdot 7 $ | 12.0.47038089498624.3 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |