Normalized defining polynomial
\( x^{12} - 4 x^{11} + 11 x^{10} - 21 x^{9} + 32 x^{8} - 40 x^{7} + 45 x^{6} - 46 x^{5} + 40 x^{4} - 26 x^{3} + 12 x^{2} - 4 x + 1 \)
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: | \(50551744653\) \(\medspace = 3^{6}\cdot 37^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(7.80\) | 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: | $3^{1/2}37^{5/6}\approx 35.10713565663159$ | ||
Ramified primes: | \(3\), \(37\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{37}a^{11}-\frac{16}{37}a^{10}+\frac{18}{37}a^{9}-\frac{15}{37}a^{8}-\frac{10}{37}a^{7}+\frac{6}{37}a^{6}+\frac{10}{37}a^{5}-\frac{18}{37}a^{4}-\frac{3}{37}a^{3}+\frac{10}{37}a^{2}+\frac{3}{37}a-\frac{3}{37}$
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)
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Torsion generator: | \( -\frac{43}{37} a^{11} + \frac{133}{37} a^{10} - \frac{330}{37} a^{9} + \frac{534}{37} a^{8} - \frac{717}{37} a^{7} + \frac{778}{37} a^{6} - \frac{837}{37} a^{5} + \frac{774}{37} a^{4} - \frac{537}{37} a^{3} + \frac{162}{37} a^{2} - \frac{18}{37} a + \frac{18}{37} \) (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)
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Fundamental units: | $\frac{48}{37}a^{11}-\frac{176}{37}a^{10}+\frac{457}{37}a^{9}-\frac{831}{37}a^{8}+\frac{1185}{37}a^{7}-\frac{1414}{37}a^{6}+\frac{1516}{37}a^{5}-\frac{1493}{37}a^{4}+\frac{1188}{37}a^{3}-\frac{630}{37}a^{2}+\frac{218}{37}a-\frac{33}{37}$, $\frac{19}{37}a^{11}-\frac{82}{37}a^{10}+\frac{231}{37}a^{9}-\frac{433}{37}a^{8}+\frac{661}{37}a^{7}-\frac{774}{37}a^{6}+\frac{856}{37}a^{5}-\frac{823}{37}a^{4}+\frac{720}{37}a^{3}-\frac{402}{37}a^{2}+\frac{94}{37}a+\frac{17}{37}$, $\frac{106}{37}a^{11}-\frac{364}{37}a^{10}+\frac{946}{37}a^{9}-\frac{1664}{37}a^{8}+\frac{2381}{37}a^{7}-\frac{2805}{37}a^{6}+\frac{3058}{37}a^{5}-\frac{3018}{37}a^{4}+\frac{2383}{37}a^{3}-\frac{1308}{37}a^{2}+\frac{503}{37}a-\frac{133}{37}$, $a^{11}-3a^{10}+8a^{9}-13a^{8}+19a^{7}-21a^{6}+24a^{5}-22a^{4}+18a^{3}-8a^{2}+3a$, $\frac{4}{37}a^{11}-\frac{64}{37}a^{10}+\frac{146}{37}a^{9}-\frac{356}{37}a^{8}+\frac{478}{37}a^{7}-\frac{642}{37}a^{6}+\frac{632}{37}a^{5}-\frac{701}{37}a^{4}+\frac{580}{37}a^{3}-\frac{330}{37}a^{2}+\frac{86}{37}a-\frac{12}{37}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 3.95685086216 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3.95685086216 \cdot 1}{6\cdot\sqrt{50551744653}}\cr\approx \mathstrut & 0.180471890197 \end{aligned}\]
Galois group
$C_6\wr C_2$ (as 12T42):
A solvable group of order 72 |
The 27 conjugacy class representatives for $C_6\wr C_2$ |
Character table for $C_6\wr C_2$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.333.1, 6.0.36963.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 24 sibling: | data not computed |
Degree 36 siblings: | data not computed |
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 | ${\href{/padicField/2.12.0.1}{12} }$ | R | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 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 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(37\) | $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | 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 | $[\ ]$ | |
37.6.5.1 | $x^{6} + 37$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |