Normalized defining polynomial
\( x^{12} - 6x^{10} + 16x^{8} - 22x^{6} + 16x^{4} - 8x^{2} + 4 \)
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: | \(7341411926016\) \(\medspace = 2^{22}\cdot 3^{6}\cdot 7^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.81\) | 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: | $2^{11/6}3^{1/2}7^{2/3}\approx 22.58643281430495$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$
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: | \( \frac{1}{2} a^{10} - 2 a^{8} + \frac{7}{2} a^{6} - 2 a^{4} \) (order $6$) | 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: | $\frac{1}{2}a^{10}-\frac{5}{2}a^{8}+\frac{11}{2}a^{6}-6a^{4}+3a^{2}-2$, $\frac{1}{2}a^{10}-2a^{8}+\frac{7}{2}a^{6}-2a^{4}+a^{2}-2$, $\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-2a^{9}-2a^{8}+\frac{7}{2}a^{7}+\frac{7}{2}a^{6}-2a^{5}-2a^{4}+a^{3}-2a-1$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{5}{2}a^{9}+2a^{8}+6a^{7}-\frac{7}{2}a^{6}-7a^{5}+2a^{4}+4a^{3}-a+1$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{5}{2}a^{9}+2a^{8}+6a^{7}-\frac{7}{2}a^{6}-7a^{5}+2a^{4}+5a^{3}-2a$ | 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: | \( 86.14899363665334 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 86.14899363665334 \cdot 1}{6\cdot\sqrt{7341411926016}}\cr\approx \mathstrut & 0.326053128620505 \end{aligned}\]
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
\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 6.0.21168.1 |
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: | 18.6.86675003008018355847168.2, 18.0.2340225081216495607873536.2 |
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 | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.22.27 | $x^{12} + 8 x^{9} + 16 x^{8} + 4 x^{7} + 2 x^{6} + 40 x^{5} + 104 x^{4} - 48 x^{3} - 12 x^{2} + 56 x + 196$ | $6$ | $2$ | $22$ | $C_6\times S_3$ | $[3]_{3}^{6}$ |
\(3\) | 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(7\) | 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.24.2t1.b.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{-6}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (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) | $0$ | $1$ | |
1.56.6t1.b.a | $1$ | $ 2^{3} \cdot 7 $ | 6.6.1229312.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.168.6t1.c.a | $1$ | $ 2^{3} \cdot 3 \cdot 7 $ | 6.0.33191424.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.21.6t1.a.b | $1$ | $ 3 \cdot 7 $ | 6.0.64827.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.168.6t1.c.b | $1$ | $ 2^{3} \cdot 3 \cdot 7 $ | 6.0.33191424.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.56.6t1.b.b | $1$ | $ 2^{3} \cdot 7 $ | 6.6.1229312.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
2.588.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 7^{2}$ | 3.1.588.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.9408.6t3.c.a | $2$ | $ 2^{6} \cdot 3 \cdot 7^{2}$ | 6.0.132765696.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.84.6t5.b.a | $2$ | $ 2^{2} \cdot 3 \cdot 7 $ | 6.0.21168.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.1344.12t18.a.a | $2$ | $ 2^{6} \cdot 3 \cdot 7 $ | 12.0.7341411926016.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.84.6t5.b.b | $2$ | $ 2^{2} \cdot 3 \cdot 7 $ | 6.0.21168.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.1344.12t18.a.b | $2$ | $ 2^{6} \cdot 3 \cdot 7 $ | 12.0.7341411926016.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |