Normalized defining polynomial
\( x^{12} + 2x^{10} + 7x^{8} + 16x^{6} - 17x^{4} - 42x^{2} - 19 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-664672764166144\) \(\medspace = -\,2^{28}\cdot 19^{5}\) | 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: | \(17.19\) | 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: | $2^{31/12}19^{1/2}\approx 26.123876537989126$ | ||
Ramified primes: | \(2\), \(19\) | 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: | \(\Q(\sqrt{-19}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{844}a^{10}+\frac{11}{211}a^{8}-\frac{11}{211}a^{6}-\frac{1}{2}a^{5}-\frac{36}{211}a^{4}+\frac{265}{844}a^{2}-\frac{1}{2}a-\frac{153}{422}$, $\frac{1}{844}a^{11}+\frac{11}{211}a^{9}-\frac{11}{211}a^{7}-\frac{36}{211}a^{5}-\frac{1}{2}a^{4}+\frac{265}{844}a^{3}-\frac{153}{422}a-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{93}{844}a^{10}+\frac{83}{844}a^{8}+\frac{275}{422}a^{6}+\frac{239}{211}a^{4}-\frac{2363}{844}a^{2}-\frac{1239}{844}$, $\frac{74}{211}a^{10}+\frac{91}{211}a^{8}+\frac{873}{422}a^{6}+\frac{1687}{422}a^{4}-\frac{4035}{422}a^{2}-\frac{3299}{422}$, $\frac{103}{844}a^{10}+\frac{101}{844}a^{8}+\frac{133}{211}a^{6}+\frac{391}{422}a^{4}-\frac{3511}{844}a^{2}-\frac{2611}{844}$, $\frac{21}{211}a^{11}-\frac{43}{844}a^{10}+\frac{109}{844}a^{9}+\frac{7}{844}a^{8}+\frac{131}{211}a^{7}-\frac{109}{422}a^{6}+\frac{493}{422}a^{5}-\frac{69}{422}a^{4}-\frac{897}{422}a^{3}+\frac{1265}{844}a^{2}-\frac{1861}{844}a+\frac{287}{844}$, $\frac{41}{844}a^{11}-\frac{101}{422}a^{10}+\frac{29}{211}a^{9}-\frac{237}{844}a^{8}+\frac{153}{422}a^{7}-\frac{310}{211}a^{6}+\frac{212}{211}a^{5}-\frac{535}{211}a^{4}-\frac{529}{844}a^{3}+\frac{1282}{211}a^{2}-\frac{1209}{422}a+\frac{3787}{844}$, $\frac{9}{844}a^{11}-\frac{89}{422}a^{10}-\frac{13}{422}a^{9}-\frac{59}{211}a^{8}+\frac{13}{422}a^{7}-\frac{515}{422}a^{6}-\frac{15}{422}a^{5}-\frac{555}{211}a^{4}-\frac{569}{844}a^{3}+\frac{1184}{211}a^{2}+\frac{733}{422}a+\frac{1168}{211}$ | 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: | \( 714.514529701 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 714.514529701 \cdot 1}{2\cdot\sqrt{664672764166144}}\cr\approx \mathstrut & 0.542795777313 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T22):
A solvable group of order 48 |
The 10 conjugacy class representatives for $C_2 \times S_4$ |
Character table for $C_2 \times S_4$ |
Intermediate fields
3.1.76.1, 6.2.369664.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.0.739328.1, 6.2.14047232.1 |
Degree 8 siblings: | 8.4.136651472896.4, 8.0.378535936.2 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.739328.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.28.94 | $x^{12} + 6 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{4} + 2$ | $12$ | $1$ | $28$ | $C_2 \times S_4$ | $[8/3, 8/3, 3]_{3}^{2}$ |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |