Normalized defining polynomial
\( x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - 4x^{7} + 7x^{6} - x^{5} - 8x^{4} + 8x^{3} - 2x^{2} - x + 1 \)
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: | \(-2928329928704\) \(\medspace = -\,2^{12}\cdot 59^{5}\) | 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: | \(10.94\) | 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: | $2^{3/2}59^{1/2}\approx 21.72556098240043$ | ||
Ramified primes: | \(2\), \(59\) | 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: | \(\Q(\sqrt{-59}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{509}a^{11}-\frac{228}{509}a^{10}-\frac{161}{509}a^{9}-\frac{102}{509}a^{8}+\frac{250}{509}a^{7}+\frac{254}{509}a^{6}-\frac{134}{509}a^{5}-\frac{123}{509}a^{4}-\frac{82}{509}a^{3}-\frac{211}{509}a^{2}+\frac{49}{509}a+\frac{74}{509}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | $\frac{363}{509}a^{11}-\frac{306}{509}a^{10}+\frac{92}{509}a^{9}-\frac{378}{509}a^{8}+\frac{148}{509}a^{7}-\frac{1454}{509}a^{6}+\frac{2258}{509}a^{5}+\frac{652}{509}a^{4}-\frac{3298}{509}a^{3}+\frac{1284}{509}a^{2}+\frac{990}{509}a-\frac{624}{509}$, $a$, $\frac{119}{509}a^{11}-\frac{155}{509}a^{10}+\frac{183}{509}a^{9}+\frac{78}{509}a^{8}+\frac{228}{509}a^{7}-\frac{314}{509}a^{6}+\frac{851}{509}a^{5}-\frac{385}{509}a^{4}-\frac{1614}{509}a^{3}+\frac{1868}{509}a^{2}+\frac{232}{509}a-\frac{865}{509}$, $\frac{9}{509}a^{11}-\frac{16}{509}a^{10}+\frac{78}{509}a^{9}+\frac{100}{509}a^{8}+\frac{214}{509}a^{7}+\frac{250}{509}a^{6}+\frac{321}{509}a^{5}-\frac{89}{509}a^{4}-\frac{229}{509}a^{3}+\frac{137}{509}a^{2}-\frac{577}{509}a+\frac{157}{509}$, $\frac{452}{509}a^{11}-\frac{238}{509}a^{10}+\frac{524}{509}a^{9}-\frac{294}{509}a^{8}+\frac{511}{509}a^{7}-\frac{1753}{509}a^{6}+\frac{2548}{509}a^{5}-\frac{115}{509}a^{4}-\frac{2452}{509}a^{3}+\frac{2356}{509}a^{2}-\frac{757}{509}a-\frac{146}{509}$, $\frac{166}{509}a^{11}-\frac{182}{509}a^{10}+\frac{251}{509}a^{9}-\frac{135}{509}a^{8}+\frac{271}{509}a^{7}-\frac{592}{509}a^{6}+\frac{1170}{509}a^{5}-\frac{567}{509}a^{4}-\frac{1396}{509}a^{3}+\frac{1622}{509}a^{2}-\frac{519}{509}a+\frac{68}{509}$ | 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: | \( 19.2003177427 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 19.2003177427 \cdot 1}{2\cdot\sqrt{2928329928704}}\cr\approx \mathstrut & 0.219749426430 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T22):
A solvable group of order 48 |
The 10 conjugacy class representatives for $C_2 \times S_4$ |
Character table for $C_2 \times S_4$ |
Intermediate fields
3.1.59.1, 6.2.222784.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.1643032.1, 6.0.27848.1 |
Degree 8 siblings: | 8.0.14258176.2, 8.4.49632710656.3 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.27848.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | R |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
2.8.12.2 | $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
\(59\) | 59.2.1.1 | $x^{2} + 118$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |