Normalized defining polynomial
\( x^{12} - 2 x^{11} - 6 x^{10} + 30 x^{9} - 17 x^{8} - 50 x^{7} + 172 x^{6} - 204 x^{5} + 266 x^{4} + \cdots + 125 \)
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: | \(2779905883635712\) \(\medspace = 2^{18}\cdot 13^{9}\) | 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: | \(19.36\) | 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: | \(\Q(\sqrt{13}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{265}a^{9}-\frac{4}{53}a^{8}+\frac{128}{265}a^{7}-\frac{112}{265}a^{6}+\frac{91}{265}a^{5}+\frac{10}{53}a^{4}+\frac{62}{265}a^{3}-\frac{93}{265}a^{2}-\frac{122}{265}a+\frac{3}{53}$, $\frac{1}{265}a^{10}-\frac{7}{265}a^{8}+\frac{63}{265}a^{7}-\frac{29}{265}a^{6}+\frac{3}{53}a^{5}+\frac{2}{265}a^{4}+\frac{87}{265}a^{3}-\frac{127}{265}a^{2}-\frac{8}{53}a+\frac{7}{53}$, $\frac{1}{1783504325}a^{11}-\frac{468629}{356700865}a^{10}-\frac{2448941}{1783504325}a^{9}+\frac{234624333}{1783504325}a^{8}-\frac{639305346}{1783504325}a^{7}+\frac{573936963}{1783504325}a^{6}+\frac{633973138}{1783504325}a^{5}+\frac{137766927}{1783504325}a^{4}+\frac{112233513}{356700865}a^{3}+\frac{168916367}{1783504325}a^{2}+\frac{709522173}{1783504325}a+\frac{17909263}{71340173}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $5$ |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{134877}{16362425} a^{11} + \frac{40892}{3272485} a^{10} + \frac{937757}{16362425} a^{9} - \frac{3610431}{16362425} a^{8} + \frac{515152}{16362425} a^{7} + \frac{7421294}{16362425} a^{6} - \frac{20663451}{16362425} a^{5} + \frac{19764911}{16362425} a^{4} - \frac{3928168}{3272485} a^{3} + \frac{10675001}{16362425} a^{2} - \frac{31607796}{16362425} a + \frac{374126}{654497} \) (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: | $\frac{3235298}{1783504325}a^{11}+\frac{377444}{71340173}a^{10}-\frac{27766448}{1783504325}a^{9}-\frac{22949586}{1783504325}a^{8}+\frac{239953082}{1783504325}a^{7}+\frac{104239644}{1783504325}a^{6}-\frac{41196206}{1783504325}a^{5}+\frac{817777791}{1783504325}a^{4}+\frac{102351406}{356700865}a^{3}+\frac{953806286}{1783504325}a^{2}-\frac{296396436}{1783504325}a+\frac{50539201}{71340173}$, $\frac{3235298}{1783504325}a^{11}+\frac{377444}{71340173}a^{10}-\frac{27766448}{1783504325}a^{9}-\frac{22949586}{1783504325}a^{8}+\frac{239953082}{1783504325}a^{7}+\frac{104239644}{1783504325}a^{6}-\frac{41196206}{1783504325}a^{5}+\frac{817777791}{1783504325}a^{4}+\frac{102351406}{356700865}a^{3}+\frac{953806286}{1783504325}a^{2}-\frac{296396436}{1783504325}a+\frac{121879374}{71340173}$, $\frac{7371528}{1783504325}a^{11}+\frac{1478313}{356700865}a^{10}-\frac{68852193}{1783504325}a^{9}+\frac{48734124}{1783504325}a^{8}+\frac{355820777}{1783504325}a^{7}-\frac{136175871}{1783504325}a^{6}+\frac{374412319}{1783504325}a^{5}+\frac{226232756}{1783504325}a^{4}-\frac{199912374}{356700865}a^{3}+\frac{3223254311}{1783504325}a^{2}-\frac{1110512516}{1783504325}a+\frac{46905812}{71340173}$, $\frac{1188772}{356700865}a^{11}-\frac{2919606}{356700865}a^{10}-\frac{10977529}{356700865}a^{9}+\frac{42501138}{356700865}a^{8}+\frac{9803799}{356700865}a^{7}-\frac{144624356}{356700865}a^{6}+\frac{73051449}{356700865}a^{5}-\frac{45759758}{356700865}a^{4}+\frac{355302844}{356700865}a^{3}-\frac{334250688}{356700865}a^{2}-\frac{15497005}{71340173}a-\frac{61591991}{71340173}$, $\frac{20547937}{1783504325}a^{11}-\frac{1328644}{71340173}a^{10}-\frac{127115797}{1783504325}a^{9}+\frac{580691541}{1783504325}a^{8}-\frac{183361022}{1783504325}a^{7}-\frac{1096654069}{1783504325}a^{6}+\frac{3392747126}{1783504325}a^{5}-\frac{2397767696}{1783504325}a^{4}+\frac{203377326}{71340173}a^{3}-\frac{3332444486}{1783504325}a^{2}+\frac{8368553336}{1783504325}a-\frac{78093671}{71340173}$ | 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: | \( 919.513377747 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 919.513377747 \cdot 3}{4\cdot\sqrt{2779905883635712}}\cr\approx \mathstrut & 0.804791682354 \end{aligned}\]
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
\(\Q(\sqrt{-1}) \), 3.3.169.1, 4.0.832.1, 6.0.1827904.1 |
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: | 12.6.564668382613504.4 |
Minimal sibling: | 12.6.564668382613504.4 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{6}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.12.0.1}{12} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.1.0.1}{1} }^{12}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.63 | $x^{12} + 6 x^{11} + 12 x^{10} + 8 x^{9} + 6 x^{8} + 24 x^{7} + 24 x^{6} + 28 x^{4} + 56 x^{3} - 24$ | $4$ | $3$ | $18$ | $D_4 \times C_3$ | $[2, 2]^{6}$ |
\(13\) | 13.6.5.5 | $x^{6} + 65$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |