Normalized defining polynomial
\( x^{12} + 204x^{10} + 15317x^{8} + 510952x^{6} + 7182806x^{4} + 34076568x^{2} + 24137569 \)
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)
| |
Discriminant: | \(114391518299003180548096\) \(\medspace = 2^{24}\cdot 7^{10}\cdot 17^{6}\) | 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: | \(83.47\) | 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))
| |
Galois root discriminant: | $2^{2}7^{5/6}17^{1/2}\approx 83.47046227189435$ | ||
Ramified primes: | \(2\), \(7\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(952=2^{3}\cdot 7\cdot 17\) | ||
Dirichlet character group: | $\lbrace$$\chi_{952}(1,·)$, $\chi_{952}(67,·)$, $\chi_{952}(101,·)$, $\chi_{952}(647,·)$, $\chi_{952}(137,·)$, $\chi_{952}(103,·)$, $\chi_{952}(237,·)$, $\chi_{952}(783,·)$, $\chi_{952}(611,·)$, $\chi_{952}(681,·)$, $\chi_{952}(883,·)$, $\chi_{952}(509,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{-238}) \), 6.0.6039609856.5$^{3}$, 6.0.42277268992.6$^{3}$, 12.0.114391518299003180548096.2$^{24}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{17}a^{2}$, $\frac{1}{17}a^{3}$, $\frac{1}{289}a^{4}$, $\frac{1}{289}a^{5}$, $\frac{1}{4913}a^{6}$, $\frac{1}{4913}a^{7}$, $\frac{1}{83521}a^{8}$, $\frac{1}{83521}a^{9}$, $\frac{1}{1419857}a^{10}$, $\frac{1}{1419857}a^{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}\times C_{3612}$, which has order $14448$ (assuming GRH)
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{1}{83521}a^{8}+\frac{8}{4913}a^{6}+\frac{20}{289}a^{4}+\frac{16}{17}a^{2}+2$, $\frac{1}{289}a^{4}+\frac{4}{17}a^{2}+2$, $\frac{1}{4913}a^{6}+\frac{7}{289}a^{4}+\frac{13}{17}a^{2}+4$, $\frac{1}{4913}a^{6}+\frac{5}{289}a^{4}+\frac{5}{17}a^{2}+1$, $\frac{1}{83521}a^{8}+\frac{9}{4913}a^{6}+\frac{25}{289}a^{4}+\frac{21}{17}a^{2}+3$ (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)]
| |
Regulator: | \( 246.50546308257188 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 246.50546308257188 \cdot 14448}{2\cdot\sqrt{114391518299003180548096}}\cr\approx \mathstrut & 0.323956414700620 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_6$ (as 12T2):
An abelian group of order 12 |
The 12 conjugacy class representatives for $C_6\times C_2$ |
Character table for $C_6\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-34}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-238}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{7}, \sqrt{-34})\), 6.0.6039609856.5, \(\Q(\zeta_{28})^+\), 6.0.42277268992.6 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.1.0.1}{1} }^{12}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.24.79 | $x^{12} - 8 x^{11} + 14 x^{10} + 76 x^{9} + 138 x^{8} + 432 x^{7} + 688 x^{6} + 992 x^{5} + 1748 x^{4} + 1728 x^{3} + 1848 x^{2} + 1648 x + 968$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ |
\(7\) | 7.6.5.2 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
7.6.5.2 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(17\) | 17.12.6.1 | $x^{12} + 918 x^{11} + 351241 x^{10} + 71712630 x^{9} + 8244584136 x^{8} + 506874732756 x^{7} + 13125344775560 x^{6} + 9625198256031 x^{5} + 28457943732288 x^{4} + 16844354225613 x^{3} + 132306217741765 x^{2} + 68598705820311 x + 44162739951115$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |