Normalized defining polynomial
\( x^{12} - 2x^{11} + 3x^{10} - 2x^{9} + 2x^{8} + 14x^{7} - 19x^{6} + 18x^{5} + 2x^{4} + 18x^{3} + 7x^{2} + 2x - 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-55351463182336\) \(\medspace = -\,2^{16}\cdot 61^{5}\) | 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: | \(13.97\) | 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: | $2^{3/2}61^{1/2}\approx 22.090722034374522$ | ||
Ramified primes: | \(2\), \(61\) | 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: | \(\Q(\sqrt{-61}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{185762}a^{11}-\frac{8095}{185762}a^{10}+\frac{31733}{185762}a^{9}+\frac{397}{92881}a^{8}-\frac{17051}{185762}a^{7}-\frac{27409}{185762}a^{6}-\frac{71691}{185762}a^{5}-\frac{16163}{92881}a^{4}-\frac{31457}{185762}a^{3}+\frac{87579}{185762}a^{2}-\frac{1929}{185762}a-\frac{42745}{92881}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{58977}{185762}a^{11}-\frac{51678}{92881}a^{10}+\frac{150753}{185762}a^{9}-\frac{77167}{185762}a^{8}+\frac{98641}{185762}a^{7}+\frac{418130}{92881}a^{6}-\frac{920035}{185762}a^{5}+\frac{820833}{185762}a^{4}+\frac{337129}{185762}a^{3}+\frac{620863}{92881}a^{2}+\frac{476997}{185762}a+\frac{101355}{185762}$, $\frac{22245}{185762}a^{11}-\frac{69897}{185762}a^{10}+\frac{48933}{92881}a^{9}-\frac{77741}{185762}a^{8}+\frac{26509}{185762}a^{7}+\frac{329203}{185762}a^{6}-\frac{417727}{92881}a^{5}+\frac{642999}{185762}a^{4}+\frac{4489}{185762}a^{3}-\frac{77001}{185762}a^{2}-\frac{46232}{92881}a-\frac{358099}{185762}$, $\frac{43005}{185762}a^{11}-\frac{50184}{92881}a^{10}+\frac{81447}{92881}a^{9}-\frac{127119}{185762}a^{8}+\frac{110121}{185762}a^{7}+\frac{293006}{92881}a^{6}-\frac{500616}{92881}a^{5}+\frac{1084869}{185762}a^{4}-\frac{89401}{185762}a^{3}+\frac{330256}{92881}a^{2}+\frac{85925}{92881}a+\frac{196935}{185762}$, $\frac{25407}{185762}a^{11}-\frac{31131}{185762}a^{10}+\frac{33251}{185762}a^{9}+\frac{17981}{185762}a^{8}-\frac{17773}{185762}a^{7}+\frac{412799}{185762}a^{6}-\frac{242589}{185762}a^{5}+\frac{40001}{185762}a^{4}+\frac{477411}{185762}a^{3}+\frac{248179}{185762}a^{2}+\frac{402589}{185762}a-\frac{22245}{185762}$, $\frac{58977}{185762}a^{11}-\frac{51678}{92881}a^{10}+\frac{150753}{185762}a^{9}-\frac{77167}{185762}a^{8}+\frac{98641}{185762}a^{7}+\frac{418130}{92881}a^{6}-\frac{920035}{185762}a^{5}+\frac{820833}{185762}a^{4}+\frac{337129}{185762}a^{3}+\frac{620863}{92881}a^{2}+\frac{662759}{185762}a+\frac{287117}{185762}$, $\frac{14149}{92881}a^{11}-\frac{13882}{92881}a^{10}+\frac{3463}{92881}a^{9}+\frac{84291}{185762}a^{8}-\frac{42642}{92881}a^{7}+\frac{246877}{92881}a^{6}-\frac{95439}{92881}a^{5}-\frac{347703}{185762}a^{4}+\frac{557945}{92881}a^{3}+\frac{29850}{92881}a^{2}+\frac{199355}{92881}a+\frac{75387}{185762}$ | 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: | \( 123.737535189 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 123.737535189 \cdot 1}{2\cdot\sqrt{55351463182336}}\cr\approx \mathstrut & 0.325736364814 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
3.1.244.1, 4.2.976.1 x2, 6.2.238144.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 4 sibling: | 4.2.976.1 |
Degree 6 siblings: | 6.2.238144.1, 6.0.58107136.1 |
Degree 8 sibling: | 8.0.56712564736.2 |
Degree 12 sibling: | 12.0.3376439254122496.4 |
Minimal sibling: | 4.2.976.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.1.0.1}{1} }^{12}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.4 | $x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
2.8.12.14 | $x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$ | $4$ | $2$ | $12$ | $D_4$ | $[2, 2]^{2}$ | |
\(61\) | 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |