Normalized defining polynomial
\( x^{12} - 10x^{10} + 50x^{8} - 116x^{6} + 144x^{4} - 48x^{2} + 8 \)
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: | \(773738492592128\) \(\medspace = 2^{27}\cdot 7^{8}\) | 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: | \(17.41\) | 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: | $2^{11/4}7^{2/3}\approx 24.616776431006123$ | ||
Ramified primes: | \(2\), \(7\) | 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: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{1348}a^{10}+\frac{27}{1348}a^{8}+\frac{19}{674}a^{6}-\frac{29}{674}a^{4}+\frac{5}{337}a^{2}-\frac{164}{337}$, $\frac{1}{1348}a^{11}+\frac{27}{1348}a^{9}+\frac{19}{674}a^{7}-\frac{29}{674}a^{5}+\frac{5}{337}a^{3}-\frac{164}{337}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{83}{1348} a^{10} - \frac{198}{337} a^{8} + \frac{957}{337} a^{6} - \frac{2046}{337} a^{4} + \frac{2437}{337} a^{2} - \frac{469}{337} \) (order $4$) | 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: | $\frac{13}{674}a^{10}-\frac{309}{1348}a^{8}+\frac{831}{674}a^{6}-\frac{1051}{337}a^{4}+\frac{1141}{337}a^{2}-\frac{220}{337}$, $\frac{57}{1348}a^{10}-\frac{483}{1348}a^{8}+\frac{1083}{674}a^{6}-\frac{995}{337}a^{4}+\frac{1296}{337}a^{2}-\frac{249}{337}$, $\frac{173}{1348}a^{11}+\frac{83}{1348}a^{10}-\frac{433}{337}a^{9}-\frac{198}{337}a^{8}+\frac{2149}{337}a^{7}+\frac{957}{337}a^{6}-\frac{9735}{674}a^{5}-\frac{2046}{337}a^{4}+\frac{5583}{337}a^{3}+\frac{2437}{337}a^{2}-\frac{1075}{337}a-\frac{469}{337}$, $\frac{263}{1348}a^{11}+\frac{69}{1348}a^{10}-\frac{668}{337}a^{9}-\frac{833}{1348}a^{8}+\frac{3341}{337}a^{7}+\frac{1161}{337}a^{6}-\frac{7689}{337}a^{5}-\frac{3191}{337}a^{4}+\frac{8729}{337}a^{3}+\frac{4052}{337}a^{2}-\frac{1344}{337}a-\frac{869}{337}$, $\frac{73}{337}a^{11}+\frac{59}{674}a^{10}-\frac{725}{337}a^{9}-\frac{1195}{1348}a^{8}+\frac{7233}{674}a^{7}+\frac{1458}{337}a^{6}-\frac{8278}{337}a^{5}-\frac{6455}{674}a^{4}+\frac{9885}{337}a^{3}+\frac{3623}{337}a^{2}-\frac{2056}{337}a-\frac{1154}{337}$ (assuming GRH) | 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: | \( 676.545557853 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 676.545557853 \cdot 1}{4\cdot\sqrt{773738492592128}}\cr\approx \mathstrut & 0.374127097041 \end{aligned}\] (assuming GRH)
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
\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{7})^+\), 4.0.512.1, 6.0.153664.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 12 sibling: | 12.6.6189907940737024.2 |
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 | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.27.303 | $x^{12} + 6 x^{10} + 8 x^{9} + 570 x^{8} + 256 x^{7} + 4192 x^{6} + 8832 x^{5} + 9540 x^{4} + 3072 x^{3} + 600 x^{2} + 800 x + 1000$ | $4$ | $3$ | $27$ | $D_4 \times C_3$ | $[2, 3, 7/2]^{3}$ |
\(7\) | 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |