Normalized defining polynomial
\( x^{12} + 6x^{8} - 16x^{6} + 21x^{4} + 4x^{2} + 1 \)
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)
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Discriminant: | \(31443203915776\) \(\medspace = 2^{24}\cdot 37^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.33\) | 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))
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Galois root discriminant: | $2^{2}37^{2/3}\approx 44.41480987434714$ | ||
Ramified primes: | \(2\), \(37\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{1705}a^{10}-\frac{843}{1705}a^{8}-\frac{6}{31}a^{6}+\frac{259}{1705}a^{4}-\frac{76}{1705}a^{2}-\frac{718}{1705}$, $\frac{1}{1705}a^{11}-\frac{843}{1705}a^{9}-\frac{6}{31}a^{7}+\frac{259}{1705}a^{5}-\frac{76}{1705}a^{3}-\frac{718}{1705}a$
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: | \( -\frac{71}{1705} a^{11} + \frac{178}{1705} a^{9} - \frac{8}{31} a^{7} + \frac{2071}{1705} a^{5} - \frac{4834}{1705} a^{3} + \frac{3238}{1705} a \) (order $8$) | 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: | $\frac{72}{341}a^{11}+\frac{2}{341}a^{9}+\frac{41}{31}a^{7}-\frac{1130}{341}a^{5}+\frac{1689}{341}a^{3}+\frac{136}{341}a$, $\frac{71}{1705}a^{11}-\frac{178}{1705}a^{9}+\frac{8}{31}a^{7}-\frac{2071}{1705}a^{5}+\frac{4834}{1705}a^{3}-\frac{1533}{1705}a$, $\frac{71}{1705}a^{11}-\frac{72}{341}a^{10}-\frac{178}{1705}a^{9}-\frac{2}{341}a^{8}+\frac{8}{31}a^{7}-\frac{41}{31}a^{6}-\frac{2071}{1705}a^{5}+\frac{1130}{341}a^{4}+\frac{4834}{1705}a^{3}-\frac{1689}{341}a^{2}-\frac{3238}{1705}a-\frac{477}{341}$, $\frac{269}{341}a^{11}-\frac{797}{1705}a^{10}-\frac{2}{341}a^{9}+\frac{101}{1705}a^{8}+\frac{145}{31}a^{7}-\frac{85}{31}a^{6}-\frac{4326}{341}a^{5}+\frac{13522}{1705}a^{4}+\frac{5472}{341}a^{3}-\frac{17858}{1705}a^{2}+\frac{1228}{341}a-\frac{634}{1705}$, $\frac{66}{155}a^{11}+\frac{437}{1705}a^{10}+\frac{7}{155}a^{9}-\frac{111}{1705}a^{8}+\frac{77}{31}a^{7}+\frac{44}{31}a^{6}-\frac{1041}{155}a^{5}-\frac{7872}{1705}a^{4}+\frac{1184}{155}a^{3}+\frac{9413}{1705}a^{2}+\frac{352}{155}a-\frac{46}{1705}$ | 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: | \( 213.51678526536452 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 213.51678526536452 \cdot 1}{8\cdot\sqrt{31443203915776}}\cr\approx \mathstrut & 0.292858510433397 \end{aligned}\]
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
\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{8})\), 6.0.87616.1 |
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: | 18.6.56547127152441105078762340352.1, 18.0.56547127152441105078762340352.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.24.79 | $x^{12} - 8 x^{11} + 14 x^{10} + 76 x^{9} + 138 x^{8} + 432 x^{7} + 688 x^{6} + 992 x^{5} + 1748 x^{4} + 1728 x^{3} + 1848 x^{2} + 1648 x + 968$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ |
\(37\) | 37.6.4.3 | $x^{6} - 1221 x^{3} + 2738$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
37.6.0.1 | $x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.148.6t1.b.a | $1$ | $ 2^{2} \cdot 37 $ | 6.0.119946304.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.37.3t1.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) | $0$ | $1$ | |
1.296.6t1.d.a | $1$ | $ 2^{3} \cdot 37 $ | 6.6.959570432.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.296.6t1.a.a | $1$ | $ 2^{3} \cdot 37 $ | 6.0.959570432.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.148.6t1.b.b | $1$ | $ 2^{2} \cdot 37 $ | 6.0.119946304.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.296.6t1.a.b | $1$ | $ 2^{3} \cdot 37 $ | 6.0.959570432.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.296.6t1.d.b | $1$ | $ 2^{3} \cdot 37 $ | 6.6.959570432.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
2.5476.3t2.a.a | $2$ | $ 2^{2} \cdot 37^{2}$ | 3.1.5476.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.87616.6t3.b.a | $2$ | $ 2^{6} \cdot 37^{2}$ | 6.0.3838281728.2 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.148.6t5.b.a | $2$ | $ 2^{2} \cdot 37 $ | 6.0.87616.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.2368.12t18.b.a | $2$ | $ 2^{6} \cdot 37 $ | 12.0.31443203915776.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.148.6t5.b.b | $2$ | $ 2^{2} \cdot 37 $ | 6.0.87616.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.2368.12t18.b.b | $2$ | $ 2^{6} \cdot 37 $ | 12.0.31443203915776.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |