Normalized defining polynomial
\( x^{12} - 2x^{10} + 9x^{8} - 9x^{6} + 20x^{4} - 7x^{2} + 1 \)
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: | \(5596214759424\) \(\medspace = 2^{12}\cdot 3^{6}\cdot 37^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.54\) | 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: | $2\cdot 3^{1/2}37^{2/3}\approx 38.464353655440554$ | ||
Ramified primes: | \(2\), \(3\), \(37\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{77}a^{10}+\frac{4}{77}a^{8}+\frac{3}{7}a^{6}+\frac{5}{11}a^{4}-\frac{1}{77}a^{2}-\frac{13}{77}$, $\frac{1}{77}a^{11}+\frac{4}{77}a^{9}+\frac{3}{7}a^{7}+\frac{5}{11}a^{5}-\frac{1}{77}a^{3}-\frac{13}{77}a$
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: | \( \frac{51}{77} a^{11} - \frac{104}{77} a^{9} + \frac{41}{7} a^{7} - \frac{64}{11} a^{5} + \frac{950}{77} a^{3} - \frac{278}{77} a \) (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: | $\frac{26}{77}a^{10}-\frac{50}{77}a^{8}+\frac{22}{7}a^{6}-\frac{35}{11}a^{4}+\frac{590}{77}a^{2}-\frac{184}{77}$, $\frac{15}{77}a^{10}-\frac{17}{77}a^{8}+\frac{10}{7}a^{6}-\frac{2}{11}a^{4}+\frac{216}{77}a^{2}+\frac{113}{77}$, $\frac{51}{77}a^{11}-\frac{104}{77}a^{9}+\frac{41}{7}a^{7}-\frac{64}{11}a^{5}+\frac{950}{77}a^{3}-\frac{278}{77}a-1$, $\frac{39}{77}a^{11}-\frac{26}{77}a^{10}-\frac{75}{77}a^{9}+\frac{50}{77}a^{8}+\frac{33}{7}a^{7}-\frac{22}{7}a^{6}-\frac{47}{11}a^{5}+\frac{35}{11}a^{4}+\frac{808}{77}a^{3}-\frac{590}{77}a^{2}-\frac{122}{77}a+\frac{107}{77}$, $\frac{79}{77}a^{11}-\frac{40}{77}a^{10}-\frac{146}{77}a^{9}+\frac{71}{77}a^{8}+\frac{62}{7}a^{7}-\frac{29}{7}a^{6}-\frac{89}{11}a^{5}+\frac{42}{11}a^{4}+\frac{1461}{77}a^{3}-\frac{653}{77}a^{2}-\frac{411}{77}a+\frac{212}{77}$ | 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: | \( 123.785869620809 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 123.785869620809 \cdot 1}{12\cdot\sqrt{5596214759424}}\cr\approx \mathstrut & 0.268300727237975 \end{aligned}\]
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
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 6.0.36963.1 |
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: | 18.6.33966586417892403297890598912.1, 18.0.1258021719181200122144096256.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
\(3\) | 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(37\) | $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
37.3.2.1 | $x^{3} + 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
37.3.2.1 | $x^{3} + 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.148.6t1.b.a | $1$ | $ 2^{2} \cdot 37 $ | 6.0.119946304.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.444.6t1.b.a | $1$ | $ 2^{2} \cdot 3 \cdot 37 $ | 6.6.3238550208.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.444.6t1.b.b | $1$ | $ 2^{2} \cdot 3 \cdot 37 $ | 6.6.3238550208.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.111.6t1.b.a | $1$ | $ 3 \cdot 37 $ | 6.0.50602347.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.111.6t1.b.b | $1$ | $ 3 \cdot 37 $ | 6.0.50602347.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.148.6t1.b.b | $1$ | $ 2^{2} \cdot 37 $ | 6.0.119946304.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.37.3t1.a.a | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.37.3t1.a.b | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
2.4107.3t2.a.a | $2$ | $ 3 \cdot 37^{2}$ | 3.1.4107.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.65712.6t3.b.a | $2$ | $ 2^{4} \cdot 3 \cdot 37^{2}$ | 6.2.3238550208.6 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.111.6t5.a.a | $2$ | $ 3 \cdot 37 $ | 6.0.36963.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.1776.12t18.b.a | $2$ | $ 2^{4} \cdot 3 \cdot 37 $ | 12.0.5596214759424.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.111.6t5.a.b | $2$ | $ 3 \cdot 37 $ | 6.0.36963.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.1776.12t18.b.b | $2$ | $ 2^{4} \cdot 3 \cdot 37 $ | 12.0.5596214759424.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |