Normalized defining polynomial
\( x^{12} - 6x^{10} + 16x^{8} - 26x^{6} + 28x^{4} - 16x^{2} + 4 \)
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: | \( -\frac{1}{2} a^{10} + 3 a^{8} - \frac{15}{2} a^{6} + 11 a^{4} - 10 a^{2} + 4 \) (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: | $a^{2}-1$, $\frac{1}{2}a^{10}-3a^{8}+\frac{15}{2}a^{6}-11a^{4}+11a^{2}-5$, $a+1$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{5}{2}a^{9}+3a^{8}+\frac{11}{2}a^{7}-\frac{15}{2}a^{6}-7a^{5}+11a^{4}+5a^{3}-10a^{2}+3$, $\frac{1}{2}a^{11}-3a^{9}+\frac{15}{2}a^{7}-\frac{1}{2}a^{6}-11a^{5}+2a^{4}+11a^{3}-3a^{2}-5a+2$ | 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: | \( 78.0394201828844 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 78.0394201828844 \cdot 1}{6\cdot\sqrt{7341411926016}}\cr\approx \mathstrut & 0.295360352248313 \end{aligned}\]
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: | 18.6.2340225081216495607873536.2, 18.0.86675003008018355847168.1 |
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 | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.22.29 | $x^{12} - 4 x^{9} + 16 x^{8} + 28 x^{7} + 2 x^{6} + 16 x^{5} + 104 x^{4} + 120 x^{3} - 12 x^{2} + 56 x + 196$ | $6$ | $2$ | $22$ | $C_6\times S_3$ | $[3]_{3}^{6}$ |
\(3\) | 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
7.6.4.2 | $x^{6} - 42 x^{3} + 147$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.24.2t1.a.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.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.168.6t1.b.a | $1$ | $ 2^{3} \cdot 3 \cdot 7 $ | 6.6.33191424.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.56.6t1.c.a | $1$ | $ 2^{3} \cdot 7 $ | 6.0.1229312.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.21.6t1.a.a | $1$ | $ 3 \cdot 7 $ | 6.0.64827.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.168.6t1.b.b | $1$ | $ 2^{3} \cdot 3 \cdot 7 $ | 6.6.33191424.1 | $C_6$ (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) | $0$ | $1$ | |
1.21.6t1.a.b | $1$ | $ 3 \cdot 7 $ | 6.0.64827.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.56.6t1.c.b | $1$ | $ 2^{3} \cdot 7 $ | 6.0.1229312.1 | $C_6$ (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.f.a | $2$ | $ 2^{6} \cdot 3 \cdot 7^{2}$ | 6.2.132765696.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.1344.12t18.b.a | $2$ | $ 2^{6} \cdot 3 \cdot 7 $ | 12.0.7341411926016.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 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.b.b | $2$ | $ 2^{6} \cdot 3 \cdot 7 $ | 12.0.7341411926016.2 | $C_6\times S_3$ (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) | $0$ | $0$ |