Normalized defining polynomial
\( x^{12} + 2x^{10} + 8x^{8} + 12x^{6} + 22x^{4} - 11x^{2} + 49 \)
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: | \(25112847391952896\) \(\medspace = 2^{12}\cdot 19^{10}\) | 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: | \(23.26\) | 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: | $2^{3/2}19^{5/6}\approx 32.89826497785412$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}+\frac{1}{7}a^{5}-\frac{2}{7}a^{3}-\frac{2}{7}a$, $\frac{1}{7}a^{8}+\frac{1}{7}a^{6}-\frac{2}{7}a^{4}-\frac{2}{7}a^{2}$, $\frac{1}{7}a^{9}-\frac{3}{7}a^{5}+\frac{2}{7}a$, $\frac{1}{539}a^{10}-\frac{23}{539}a^{8}+\frac{4}{49}a^{6}-\frac{10}{539}a^{4}-\frac{267}{539}a^{2}+\frac{4}{11}$, $\frac{1}{539}a^{11}-\frac{23}{539}a^{9}-\frac{3}{49}a^{7}-\frac{87}{539}a^{5}-\frac{113}{539}a^{3}-\frac{27}{77}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{2}{539}a^{11}+\frac{31}{539}a^{9}+\frac{8}{49}a^{7}+\frac{288}{539}a^{5}+\frac{544}{539}a^{3}+\frac{78}{77}a$, $\frac{3}{539}a^{11}+\frac{8}{539}a^{9}+\frac{5}{49}a^{7}+\frac{201}{539}a^{5}+\frac{431}{539}a^{3}+\frac{51}{77}a$, $\frac{13}{539}a^{11}-\frac{10}{539}a^{10}+\frac{9}{539}a^{9}-\frac{1}{539}a^{8}+\frac{10}{49}a^{7}-\frac{12}{49}a^{6}+\frac{101}{539}a^{5}+\frac{23}{539}a^{4}+\frac{148}{539}a^{3}-\frac{641}{539}a^{2}+\frac{45}{77}a+\frac{4}{11}$, $\frac{5}{49}a^{11}-\frac{80}{539}a^{10}+\frac{18}{49}a^{9}-\frac{239}{539}a^{8}+\frac{59}{49}a^{7}-\frac{68}{49}a^{6}+\frac{125}{49}a^{5}-\frac{1510}{539}a^{4}+\frac{212}{49}a^{3}-\frac{1971}{539}a^{2}+a+\frac{21}{11}$, $\frac{30}{539}a^{11}-\frac{20}{539}a^{10}+\frac{3}{539}a^{9}-\frac{156}{539}a^{8}+\frac{1}{49}a^{7}-\frac{38}{49}a^{6}-\frac{454}{539}a^{5}-\frac{1263}{539}a^{4}-\frac{1080}{539}a^{3}-\frac{2052}{539}a^{2}-\frac{436}{77}a-\frac{58}{11}$ | 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: | \( 2425.67745873 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2425.67745873 \cdot 1}{2\cdot\sqrt{25112847391952896}}\cr\approx \mathstrut & 0.470906069572 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T7):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4 \times C_2$ |
Character table for $A_4 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{-19}) \), 3.3.361.1, 6.4.158470336.1, 6.0.2476099.1, 6.2.8340544.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.4.158470336.1 |
Degree 8 sibling: | 8.0.192699928576.3 |
Degree 12 sibling: | 12.4.1607222233084985344.3 |
Minimal sibling: | 6.4.158470336.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.11 | $x^{12} + 28 x^{10} + 40 x^{9} + 356 x^{8} + 896 x^{7} + 2720 x^{6} + 6656 x^{5} + 12464 x^{4} + 19456 x^{3} + 26304 x^{2} + 19840 x + 5824$ | $2$ | $6$ | $12$ | $A_4 \times C_2$ | $[2, 2]^{6}$ |
\(19\) | 19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |