Normalized defining polynomial
\( x^{12} - 13x^{10} + 22x^{8} - 104x^{6} + x^{4} - 195x^{2} + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(5881408402696281\) \(\medspace = 3^{6}\cdot 7^{10}\cdot 13^{4}\) | 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: | \(20.61\) | 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: | $3^{1/2}7^{5/6}13^{1/2}\approx 31.606810323667133$ | ||
Ramified primes: | \(3\), \(7\), \(13\) | 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 not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{26}a^{8}-\frac{1}{2}a^{5}-\frac{7}{26}a^{4}-\frac{1}{2}a^{2}+\frac{9}{26}$, $\frac{1}{26}a^{9}+\frac{3}{13}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{2}{13}a-\frac{1}{2}$, $\frac{1}{53534}a^{10}-\frac{843}{53534}a^{8}-\frac{77}{923}a^{6}-\frac{1}{2}a^{5}+\frac{6415}{26767}a^{4}-\frac{1}{2}a^{3}+\frac{8485}{53534}a^{2}-\frac{1}{2}a+\frac{23743}{53534}$, $\frac{1}{53534}a^{11}-\frac{843}{53534}a^{9}-\frac{77}{923}a^{7}-\frac{13937}{53534}a^{5}-\frac{1}{2}a^{4}+\frac{8485}{53534}a^{3}-\frac{1512}{26767}a-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $7$ | 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: | $\frac{233}{26767}a^{10}-\frac{221}{2059}a^{8}+\frac{115}{923}a^{6}-\frac{1922}{2059}a^{4}-\frac{3753}{26767}a^{2}-\frac{2566}{2059}$, $\frac{184}{26767}a^{10}-\frac{2746}{26767}a^{8}+\frac{277}{923}a^{6}-\frac{17425}{26767}a^{4}+\frac{8754}{26767}a^{2}-\frac{14899}{26767}$, $\frac{3215}{53534}a^{11}+\frac{417}{53534}a^{10}-\frac{41781}{53534}a^{9}-\frac{5619}{53534}a^{8}+\frac{2387}{1846}a^{7}+\frac{196}{923}a^{6}-\frac{316583}{53534}a^{5}-\frac{42411}{53534}a^{4}-\frac{23065}{53534}a^{3}+\frac{5001}{53534}a^{2}-\frac{561669}{53534}a+\frac{5277}{53534}$, $\frac{507}{4118}a^{11}-\frac{92}{26767}a^{10}-\frac{42725}{26767}a^{9}+\frac{1373}{26767}a^{8}+\frac{377}{142}a^{7}-\frac{277}{1846}a^{6}-\frac{682129}{53534}a^{5}+\frac{17425}{53534}a^{4}-\frac{1415}{4118}a^{3}-\frac{4377}{26767}a^{2}-\frac{1262231}{53534}a+\frac{20833}{26767}$, $\frac{507}{4118}a^{11}+\frac{25}{2059}a^{10}-\frac{42725}{26767}a^{9}-\frac{4246}{26767}a^{8}+\frac{377}{142}a^{7}+\frac{39}{142}a^{6}-\frac{682129}{53534}a^{5}-\frac{67397}{53534}a^{4}-\frac{1415}{4118}a^{3}+\frac{48}{2059}a^{2}-\frac{1262231}{53534}a-\frac{54191}{26767}$, $\frac{507}{4118}a^{11}+\frac{25}{2059}a^{10}-\frac{42725}{26767}a^{9}-\frac{4246}{26767}a^{8}+\frac{377}{142}a^{7}+\frac{39}{142}a^{6}-\frac{682129}{53534}a^{5}-\frac{67397}{53534}a^{4}-\frac{1415}{4118}a^{3}+\frac{48}{2059}a^{2}-\frac{1315765}{53534}a-\frac{54191}{26767}$, $\frac{233}{26767}a^{11}-\frac{221}{2059}a^{9}+\frac{115}{923}a^{7}-\frac{1922}{2059}a^{5}-\frac{3753}{26767}a^{3}-\frac{4625}{2059}a$ | 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: | \( 726.081873929 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 726.081873929 \cdot 2}{2\cdot\sqrt{5881408402696281}}\cr\approx \mathstrut & 0.236093691265 \end{aligned}\]
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
\(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), 6.2.76690341.1, \(\Q(\zeta_{21})^+\), 6.2.405769.1 |
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: | 6.2.76690341.1 |
Degree 8 sibling: | 8.0.272174020209.3 |
Degree 12 sibling: | 12.0.993958020055671489.6 |
Minimal sibling: | 6.2.76690341.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 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 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |