Normalized defining polynomial
\( x^{12} - x^{10} - 18x^{8} - 22x^{6} + 6x^{4} + 10x^{2} - 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-564668382613504\) \(\medspace = -\,2^{12}\cdot 13^{10}\) | 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: | \(16.96\) | 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: | $2^{3/2}13^{5/6}\approx 23.97900291135148$ | ||
Ramified primes: | \(2\), \(13\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{265}a^{10}-\frac{10}{53}a^{8}+\frac{47}{265}a^{6}+\frac{12}{53}a^{4}-\frac{19}{265}a^{2}-\frac{119}{265}$, $\frac{1}{265}a^{11}-\frac{10}{53}a^{9}+\frac{47}{265}a^{7}+\frac{12}{53}a^{5}-\frac{19}{265}a^{3}-\frac{119}{265}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{136}{265}a^{11}-\frac{35}{53}a^{9}-\frac{2353}{265}a^{7}-\frac{488}{53}a^{5}+\frac{861}{265}a^{3}+\frac{1306}{265}a$, $\frac{16}{53}a^{11}-\frac{5}{53}a^{9}-\frac{308}{53}a^{7}-\frac{524}{53}a^{5}-\frac{39}{53}a^{3}+\frac{216}{53}a$, $\frac{137}{265}a^{11}-\frac{45}{53}a^{9}-\frac{2306}{265}a^{7}-\frac{317}{53}a^{5}+\frac{1637}{265}a^{3}+\frac{657}{265}a$, $a$, $\frac{87}{265}a^{10}-\frac{22}{53}a^{8}-\frac{1476}{265}a^{6}-\frac{334}{53}a^{4}+\frac{202}{265}a^{2}+\frac{247}{265}$, $\frac{167}{265}a^{11}-\frac{39}{265}a^{10}-\frac{27}{53}a^{9}+\frac{19}{53}a^{8}-\frac{3016}{265}a^{7}+\frac{552}{265}a^{6}-\frac{858}{53}a^{5}+\frac{9}{53}a^{4}+\frac{7}{265}a^{3}-\frac{54}{265}a^{2}+\frac{1592}{265}a+\frac{401}{265}$, $\frac{167}{265}a^{11}+\frac{1}{265}a^{10}-\frac{27}{53}a^{9}-\frac{10}{53}a^{8}-\frac{3016}{265}a^{7}+\frac{47}{265}a^{6}-\frac{858}{53}a^{5}+\frac{171}{53}a^{4}+\frac{7}{265}a^{3}+\frac{776}{265}a^{2}+\frac{1592}{265}a-\frac{384}{265}$, $\frac{167}{265}a^{11}-\frac{87}{265}a^{10}-\frac{27}{53}a^{9}+\frac{22}{53}a^{8}-\frac{3016}{265}a^{7}+\frac{1476}{265}a^{6}-\frac{858}{53}a^{5}+\frac{334}{53}a^{4}+\frac{7}{265}a^{3}-\frac{202}{265}a^{2}+\frac{1592}{265}a-\frac{512}{265}$ | 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: | \( 635.636878164 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 635.636878164 \cdot 1}{2\cdot\sqrt{564668382613504}}\cr\approx \mathstrut & 0.212325391672 \end{aligned}\]
Galois group
$C_3\times D_4$ (as 12T14):
A solvable group of order 24 |
The 15 conjugacy class representatives for $D_4 \times C_3$ |
Character table for $D_4 \times C_3$ |
Intermediate fields
\(\Q(\sqrt{13}) \), 3.3.169.1, 4.2.2704.1, \(\Q(\zeta_{13})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 12 sibling: | 12.0.2779905883635712.6 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{6}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.12.0.1}{12} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.1.0.1}{1} }^{12}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.24 | $x^{12} - 10 x^{11} + 72 x^{10} - 292 x^{9} + 1028 x^{8} - 2144 x^{7} + 5280 x^{6} - 5568 x^{5} + 12400 x^{4} - 12384 x^{3} + 24832 x^{2} - 14784 x + 11200$ | $2$ | $6$ | $12$ | $D_4 \times C_3$ | $[2, 2]^{6}$ |
\(13\) | 13.12.10.1 | $x^{12} + 72 x^{11} + 2172 x^{10} + 35280 x^{9} + 328380 x^{8} + 1703232 x^{7} + 4282170 x^{6} + 3407400 x^{5} + 1340820 x^{4} + 712800 x^{3} + 3855192 x^{2} + 18082080 x + 35650393$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |