Normalized defining polynomial
\( x^{12} - 10x^{10} + 63x^{8} - 200x^{6} + 411x^{4} + 65x^{2} + 25 \)
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: | \(274941996890625\) \(\medspace = 3^{6}\cdot 5^{6}\cdot 17^{6}\) | 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: | \(15.97\) | 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: | $3^{1/2}5^{1/2}17^{1/2}\approx 15.968719422671311$ | ||
Ramified primes: | \(3\), \(5\), \(17\) | 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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{1365850}a^{10}+\frac{19439}{273170}a^{8}+\frac{2166}{11575}a^{6}-\frac{1}{2}a^{5}-\frac{38016}{136585}a^{4}-\frac{380639}{1365850}a^{2}-\frac{1}{2}a+\frac{35929}{273170}$, $\frac{1}{1365850}a^{11}+\frac{19439}{273170}a^{9}+\frac{2166}{11575}a^{7}-\frac{38016}{136585}a^{5}-\frac{1}{2}a^{4}-\frac{380639}{1365850}a^{3}-\frac{50328}{136585}a-\frac{1}{2}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $5$ |
Class group and class number
$C_{4}$, which has order $4$
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)
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Fundamental units: | $\frac{51}{2950}a^{11}-\frac{53}{295}a^{9}+\frac{57}{50}a^{7}-\frac{2217}{590}a^{5}+\frac{11743}{1475}a^{3}-\frac{228}{295}a+\frac{1}{2}$, $\frac{2747}{273170}a^{11}+\frac{137}{27317}a^{10}-\frac{2901}{27317}a^{9}-\frac{2645}{54634}a^{8}+\frac{3219}{4630}a^{7}+\frac{299}{926}a^{6}-\frac{130707}{54634}a^{5}-\frac{58753}{54634}a^{4}+\frac{1445277}{273170}a^{3}+\frac{137805}{54634}a^{2}-\frac{26996}{27317}a+\frac{24813}{54634}$, $\frac{2747}{273170}a^{11}+\frac{973}{682925}a^{10}-\frac{2901}{27317}a^{9}-\frac{5751}{273170}a^{8}+\frac{3219}{4630}a^{7}+\frac{1736}{11575}a^{6}-\frac{130707}{54634}a^{5}-\frac{86651}{136585}a^{4}+\frac{1445277}{273170}a^{3}+\frac{1149453}{682925}a^{2}-\frac{81309}{54634}a-\frac{280013}{136585}$, $\frac{18674}{682925}a^{11}-\frac{973}{682925}a^{10}-\frac{39044}{136585}a^{9}+\frac{5751}{273170}a^{8}+\frac{21243}{11575}a^{7}-\frac{1736}{11575}a^{6}-\frac{840003}{136585}a^{5}+\frac{86651}{136585}a^{4}+\frac{18100403}{1365850}a^{3}-\frac{1149453}{682925}a^{2}-\frac{617673}{273170}a+\frac{423441}{273170}$, $\frac{21667}{1365850}a^{11}+\frac{428}{27317}a^{10}-\frac{43327}{273170}a^{9}-\frac{4331}{27317}a^{8}+\frac{22919}{23150}a^{7}+\frac{951}{926}a^{6}-\frac{853169}{273170}a^{5}-\frac{90379}{27317}a^{4}+\frac{4287556}{682925}a^{3}+\frac{196315}{27317}a^{2}+\frac{174449}{136585}a+\frac{18022}{27317}$ | 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: | \( 102.376240284 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 102.376240284 \cdot 4}{2\cdot\sqrt{274941996890625}}\cr\approx \mathstrut & 0.759779971158 \end{aligned}\]
Galois group
A solvable group of order 12 |
The 6 conjugacy class representatives for $D_6$ |
Character table for $D_6$ |
Intermediate fields
\(\Q(\sqrt{-255}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-51}) \), 3.1.255.1 x3, \(\Q(\sqrt{5}, \sqrt{-51})\), 6.0.16581375.1, 6.2.325125.1 x3, 6.0.3316275.2 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.0.3316275.2, 6.2.325125.1 |
Minimal sibling: | 6.2.325125.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | R | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(17\) | 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |