Normalized defining polynomial
\( x^{12} + 1092x^{10} + 424242x^{8} + 70309512x^{6} + 4510403352x^{4} + 61163496576x^{2} + 222991914600 \)
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: | \(1320330028678597666666010640384\) \(\medspace = 2^{33}\cdot 3^{6}\cdot 7^{6}\cdot 13^{11}\) | 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: | \(323.64\) | 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: | $2^{11/4}3^{1/2}7^{1/2}13^{11/12}\approx 323.6360374197893$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(13\) | 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: | \(\Q(\sqrt{26}) \) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(4368=2^{4}\cdot 3\cdot 7\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{4368}(1,·)$, $\chi_{4368}(4033,·)$, $\chi_{4368}(2857,·)$, $\chi_{4368}(1805,·)$, $\chi_{4368}(2645,·)$, $\chi_{4368}(2477,·)$, $\chi_{4368}(3025,·)$, $\chi_{4368}(3317,·)$, $\chi_{4368}(2521,·)$, $\chi_{4368}(3865,·)$, $\chi_{4368}(125,·)$, $\chi_{4368}(629,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | 4.0.1984260096.2$^{2}$, 12.0.1320330028678597666666010640384.1$^{30}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{21}a^{2}$, $\frac{1}{21}a^{3}$, $\frac{1}{882}a^{4}$, $\frac{1}{4410}a^{5}+\frac{2}{5}a$, $\frac{1}{92610}a^{6}+\frac{2}{105}a^{2}$, $\frac{1}{92610}a^{7}+\frac{2}{105}a^{3}$, $\frac{1}{19448100}a^{8}+\frac{1}{231525}a^{6}-\frac{1}{7350}a^{4}-\frac{2}{175}a^{2}$, $\frac{1}{19448100}a^{9}+\frac{1}{231525}a^{7}+\frac{1}{11025}a^{5}-\frac{2}{175}a^{3}+\frac{2}{5}a$, $\frac{1}{1112917522500}a^{10}+\frac{209}{26498036250}a^{8}-\frac{191}{50472450}a^{6}-\frac{1936}{4291875}a^{4}-\frac{8359}{1430625}a^{2}-\frac{1236}{2725}$, $\frac{1}{1112917522500}a^{11}+\frac{209}{26498036250}a^{9}-\frac{191}{50472450}a^{7}+\frac{73}{30043125}a^{5}-\frac{8359}{1430625}a^{3}+\frac{944}{2725}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $5$ |
Class group and class number
$C_{2}\times C_{2}\times C_{4820212}$, which has order $19280848$ (assuming GRH)
Unit group
Rank: | $5$ | 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)
| |
Fundamental units: | $\frac{17}{111291752250}a^{10}+\frac{559}{3533071500}a^{8}+\frac{1423}{25236225}a^{6}+\frac{1981}{245250}a^{4}+\frac{37518}{95375}a^{2}+\frac{1031}{545}$, $\frac{13}{55645876125}a^{10}+\frac{887}{3533071500}a^{8}+\frac{484}{5047245}a^{6}+\frac{93721}{6008625}a^{4}+\frac{273832}{286125}a^{2}+\frac{4398}{545}$, $\frac{257}{556458761250}a^{10}+\frac{26827}{52996072500}a^{8}+\frac{79}{400575}a^{6}+\frac{980372}{30043125}a^{4}+\frac{2921999}{1430625}a^{2}+\frac{56846}{2725}$, $\frac{491}{556458761250}a^{10}+\frac{12694}{13249018125}a^{8}+\frac{2651}{7210350}a^{6}+\frac{3552347}{60086250}a^{4}+\frac{4945037}{1430625}a^{2}+\frac{56098}{2725}$, $\frac{127}{1112917522500}a^{10}+\frac{6761}{52996072500}a^{8}+\frac{2557}{50472450}a^{6}+\frac{56863}{6676250}a^{4}+\frac{106051}{204375}a^{2}+\frac{9253}{2725}$ (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: | \( 4543.270357084286 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 4543.270357084286 \cdot 19280848}{2\cdot\sqrt{1320330028678597666666010640384}}\cr\approx \mathstrut & 2.34532307370932 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 12 |
The 12 conjugacy class representatives for $C_{12}$ |
Character table for $C_{12}$ |
Intermediate fields
\(\Q(\sqrt{26}) \), 3.3.169.1, 4.0.1984260096.2, 6.6.190102016.1 |
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 | R | ${\href{/padicField/5.1.0.1}{1} }^{12}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.33.340 | $x^{12} + 76 x^{10} - 384 x^{9} - 1170 x^{8} - 512 x^{7} - 592 x^{6} + 256 x^{5} + 1484 x^{4} + 1536 x^{3} + 2992 x^{2} + 6376$ | $4$ | $3$ | $33$ | $C_{12}$ | $[3, 4]^{3}$ |
\(3\) | 3.12.6.1 | $x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
\(7\) | 7.12.6.2 | $x^{12} + 49 x^{8} - 1715 x^{6} + 9604 x^{4} - 100842 x^{2} + 352947$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
\(13\) | 13.12.11.8 | $x^{12} + 104$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |