Normalized defining polynomial
\( x^{12} - x^{11} + x^{10} - 3x^{9} + 4x^{8} - 2x^{7} + x^{6} - 2x^{5} + 4x^{4} - 3x^{3} + x^{2} - x + 1 \)
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: | \(53440955929\) \(\medspace = 19^{2}\cdot 23^{6}\) | 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: | \(7.83\) | 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: | $19^{1/2}23^{1/2}\approx 20.904544960366874$ | ||
Ramified primes: | \(19\), \(23\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$
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: | $\frac{7}{5}a^{11}+\frac{6}{5}a^{10}+\frac{11}{5}a^{9}-\frac{7}{5}a^{8}+a^{7}+\frac{1}{5}a^{6}+\frac{6}{5}a^{5}-2a^{4}+\frac{13}{5}a^{3}+\frac{6}{5}a^{2}+\frac{11}{5}a-\frac{3}{5}$, $\frac{4}{5}a^{11}+\frac{7}{5}a^{10}+\frac{12}{5}a^{9}+\frac{1}{5}a^{8}-\frac{3}{5}a^{6}+\frac{7}{5}a^{5}+\frac{1}{5}a^{3}+\frac{7}{5}a^{2}+\frac{12}{5}a+\frac{4}{5}$, $\frac{3}{5}a^{11}+\frac{4}{5}a^{10}+\frac{9}{5}a^{9}+\frac{2}{5}a^{8}+a^{7}-\frac{1}{5}a^{6}+\frac{4}{5}a^{5}+\frac{2}{5}a^{3}+\frac{4}{5}a^{2}+\frac{9}{5}a+\frac{8}{5}$, $\frac{13}{5}a^{11}-\frac{11}{5}a^{10}+\frac{4}{5}a^{9}-\frac{43}{5}a^{8}+7a^{7}-\frac{11}{5}a^{6}+\frac{4}{5}a^{5}-5a^{4}+\frac{42}{5}a^{3}-\frac{21}{5}a^{2}-\frac{1}{5}a-\frac{17}{5}$, $\frac{9}{5}a^{11}+\frac{2}{5}a^{10}+\frac{7}{5}a^{9}-\frac{19}{5}a^{8}+2a^{7}+\frac{2}{5}a^{6}+\frac{7}{5}a^{5}-3a^{4}+\frac{21}{5}a^{3}-\frac{3}{5}a^{2}+\frac{2}{5}a-\frac{11}{5}$ | 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: | \( 1.36801263468 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1.36801263468 \cdot 1}{2\cdot\sqrt{53440955929}}\cr\approx \mathstrut & 0.182054833553 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T21):
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
\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.10051.1, 6.2.231173.1, 6.0.12167.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.231173.1, 6.0.10051.1 |
Degree 8 siblings: | 8.0.68939809.1, 8.4.36469158961.1 |
Degree 12 siblings: | 12.4.6964478817623209.1, 12.2.838790656103.1, 12.0.13165366384921.1, 12.0.6964478817623209.3, 12.0.6964478817623209.1 |
Degree 16 sibling: | deg 16 |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.10051.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | R | R | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(19\) | 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(23\) | 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |