Normalized defining polynomial
\( x^{12} + 18x^{10} + 114x^{8} + 260x^{6} + 1859x^{4} - 4394x^{2} + 28561 \)
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: | \(33045608803078144\) \(\medspace = 2^{12}\cdot 7^{10}\cdot 13^{4}\) | 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: | \(23.80\) | 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: | $2^{3/2}7^{5/6}13^{1/2}\approx 51.61370512661073$ | ||
Ramified primes: | \(2\), \(7\), \(13\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})\)$^{3}$, 8.0.13763268972544.21$^{4}$, 12.0.33045608803078144.1$^{12}$, deg 24$^{12}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{26}a^{6}-\frac{4}{13}a^{4}-\frac{3}{26}a^{2}-\frac{1}{2}$, $\frac{1}{26}a^{7}-\frac{4}{13}a^{5}-\frac{3}{26}a^{3}-\frac{1}{2}a$, $\frac{1}{338}a^{8}+\frac{5}{338}a^{6}+\frac{49}{338}a^{4}+\frac{5}{13}a^{2}-\frac{1}{2}$, $\frac{1}{338}a^{9}+\frac{5}{338}a^{7}+\frac{49}{338}a^{5}+\frac{5}{13}a^{3}-\frac{1}{2}a$, $\frac{1}{131341054}a^{10}+\frac{52603}{131341054}a^{8}-\frac{2136423}{131341054}a^{6}-\frac{1232588}{5051579}a^{4}+\frac{359383}{777166}a^{2}+\frac{7615}{29891}$, $\frac{1}{131341054}a^{11}+\frac{52603}{131341054}a^{9}-\frac{2136423}{131341054}a^{7}-\frac{1232588}{5051579}a^{5}+\frac{359383}{777166}a^{3}+\frac{7615}{29891}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{4}$, which has order $8$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{1142}{65670527} a^{10} - \frac{73105}{131341054} a^{8} - \frac{506174}{65670527} a^{6} - \frac{613833}{10103158} a^{4} - \frac{113585}{777166} a^{2} + \frac{3902}{29891} \) (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: | $\frac{1219}{65670527}a^{10}+\frac{6862}{65670527}a^{8}-\frac{16371}{65670527}a^{6}-\frac{107314}{5051579}a^{4}-\frac{54401}{388583}a^{2}+\frac{3059}{29891}$, $\frac{12451}{131341054}a^{10}+\frac{98799}{65670527}a^{8}+\frac{311829}{65670527}a^{6}-\frac{183571}{10103158}a^{4}+\frac{174379}{777166}a^{2}+\frac{30117}{59782}$, $\frac{11798}{65670527}a^{11}-\frac{10656}{65670527}a^{10}+\frac{474869}{131341054}a^{9}-\frac{200882}{65670527}a^{8}+\frac{3332729}{131341054}a^{7}-\frac{2320381}{131341054}a^{6}+\frac{579507}{10103158}a^{5}+\frac{17163}{5051579}a^{4}+\frac{51957}{388583}a^{3}+\frac{9671}{777166}a^{2}-\frac{72195}{59782}a+\frac{183955}{59782}$, $\frac{3281}{65670527}a^{11}-\frac{12187}{131341054}a^{10}+\frac{59591}{65670527}a^{9}-\frac{149697}{65670527}a^{8}+\frac{1255497}{131341054}a^{7}-\frac{3136895}{131341054}a^{6}+\frac{292035}{5051579}a^{5}-\frac{792553}{10103158}a^{4}+\frac{200147}{777166}a^{3}-\frac{160378}{388583}a^{2}+\frac{13647}{59782}a+\frac{7550}{29891}$, $\frac{31479}{131341054}a^{11}-\frac{4203}{10103158}a^{10}+\frac{526257}{131341054}a^{9}-\frac{23232}{5051579}a^{8}+\frac{1931691}{131341054}a^{7}+\frac{49789}{5051579}a^{6}-\frac{415865}{5051579}a^{5}+\frac{2348435}{10103158}a^{4}+\frac{51123}{777166}a^{3}-\frac{690709}{777166}a^{2}+\frac{16656}{29891}a+\frac{103143}{59782}$ | 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: | \( 1615.30314445 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1615.30314445 \cdot 8}{14\cdot\sqrt{33045608803078144}}\cr\approx \mathstrut & 0.312419635445 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T7):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4 \times C_2$ |
Character table for $A_4 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 6.0.181784512.2, 6.6.25969216.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.0.181784512.2 |
Degree 8 sibling: | 8.0.13763268972544.21 |
Degree 12 sibling: | deg 12 |
Minimal sibling: | 6.0.181784512.2 |
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 | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.1 | $x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ |
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 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |