Normalized defining polynomial
\( x^{12} - 4x^{11} + 9x^{10} - 22x^{9} + 32x^{8} - 30x^{7} + 57x^{6} - 56x^{5} - 14x^{3} + 14x^{2} + 14x + 7 \)
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: | \(15203947802176\) \(\medspace = 2^{6}\cdot 7^{10}\cdot 29^{2}\) | 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: | \(12.55\) | 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: | $2^{3/2}7^{5/6}29^{1/2}\approx 77.08899060875882$ | ||
Ramified primes: | \(2\), \(7\), \(29\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{10532}a^{11}-\frac{34}{2633}a^{10}-\frac{235}{5266}a^{9}-\frac{587}{5266}a^{8}+\frac{1143}{5266}a^{7}-\frac{405}{2633}a^{6}-\frac{2009}{10532}a^{5}+\frac{458}{2633}a^{4}-\frac{2221}{10532}a^{3}-\frac{869}{5266}a^{2}-\frac{4907}{10532}a+\frac{5}{2633}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( \frac{997}{5266} a^{11} - \frac{5251}{10532} a^{10} + \frac{2675}{2633} a^{9} - \frac{14591}{5266} a^{8} + \frac{12129}{5266} a^{7} - \frac{7138}{2633} a^{6} + \frac{40235}{5266} a^{5} - \frac{14761}{10532} a^{4} + \frac{8}{2633} a^{3} - \frac{50571}{10532} a^{2} - \frac{5431}{5266} a - \frac{10147}{10532} \) (order $14$) | 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: | $\frac{1887}{10532}a^{11}-\frac{6497}{10532}a^{10}+\frac{3399}{2633}a^{9}-\frac{17607}{5266}a^{8}+\frac{10739}{2633}a^{7}-\frac{19761}{5266}a^{6}+\frac{95325}{10532}a^{5}-\frac{58071}{10532}a^{4}-\frac{9823}{10532}a^{3}-\frac{38383}{10532}a^{2}-\frac{1881}{10532}a+\frac{14043}{10532}$, $\frac{925}{10532}a^{11}+\frac{146}{2633}a^{10}-\frac{1469}{5266}a^{9}+\frac{1028}{2633}a^{8}-\frac{16989}{5266}a^{7}+\frac{7160}{2633}a^{6}-\frac{15225}{10532}a^{5}+\frac{54767}{5266}a^{4}-\frac{37547}{10532}a^{3}-\frac{5646}{2633}a^{2}-\frac{57609}{10532}a-\frac{14447}{5266}$, $\frac{7009}{10532}a^{11}-\frac{18509}{10532}a^{10}+\frac{9787}{2633}a^{9}-\frac{25782}{2633}a^{8}+\frac{43829}{5266}a^{7}-\frac{50569}{5266}a^{6}+\frac{258237}{10532}a^{5}-\frac{37515}{10532}a^{4}-\frac{32289}{10532}a^{3}-\frac{109337}{10532}a^{2}-\frac{79907}{10532}a-\frac{30965}{10532}$, $\frac{562}{2633}a^{11}-\frac{2783}{5266}a^{10}+\frac{6219}{5266}a^{9}-\frac{16241}{5266}a^{8}+\frac{12821}{5266}a^{7}-\frac{17275}{5266}a^{6}+\frac{18930}{2633}a^{5}+\frac{81}{2633}a^{4}-\frac{160}{2633}a^{3}-\frac{7812}{2633}a^{2}-\frac{8882}{2633}a-\frac{6483}{5266}$, $\frac{81}{2633}a^{11}-\frac{4569}{10532}a^{10}+\frac{5483}{5266}a^{9}-\frac{5572}{2633}a^{8}+\frac{25409}{5266}a^{7}-\frac{17571}{5266}a^{6}+\frac{19465}{5266}a^{5}-\frac{119975}{10532}a^{4}-\frac{857}{2633}a^{3}+\frac{60909}{10532}a^{2}+\frac{29195}{5266}a+\frac{9113}{10532}$ | 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: | \( 320.143524163 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 320.143524163 \cdot 1}{14\cdot\sqrt{15203947802176}}\cr\approx \mathstrut & 0.360842523126 \end{aligned}\]
Galois group
$C_2^3:A_4$ (as 12T58):
A solvable group of order 96 |
The 10 conjugacy class representatives for $C_2^3:A_4$ |
Character table for $C_2^3:A_4$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | data not computed |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 sibling: | data not computed |
Minimal sibling: | 8.0.6332339776.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
2.6.6.1 | $x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ | |
\(7\) | 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(29\) | $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |