Normalized defining polynomial
\( x^{12} - x^{11} - x^{10} + 2x^{9} + 11x^{8} - 11x^{7} + 2x^{6} + 19x^{5} - 19x^{4} + 14x^{3} - 3x^{2} + x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-82608050631424\) \(\medspace = -\,2^{8}\cdot 19^{9}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.45\) | 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))
| |
Galois root discriminant: | $2^{2/3}19^{3/4}\approx 14.446141500988903$ | ||
Ramified primes: | \(2\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{124}a^{10}-\frac{19}{124}a^{9}+\frac{1}{31}a^{8}+\frac{9}{124}a^{7}-\frac{11}{124}a^{6}+\frac{3}{62}a^{5}+\frac{5}{124}a^{4}+\frac{15}{124}a^{3}+\frac{5}{62}a^{2}-\frac{13}{124}a-\frac{39}{124}$, $\frac{1}{248}a^{11}+\frac{15}{248}a^{9}-\frac{39}{248}a^{8}+\frac{9}{62}a^{7}-\frac{79}{248}a^{6}-\frac{5}{248}a^{5}+\frac{55}{124}a^{4}-\frac{77}{248}a^{3}-\frac{71}{248}a^{2}+\frac{43}{124}a-\frac{121}{248}$
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)
| |
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: | $\frac{173}{248}a^{11}-\frac{7}{62}a^{10}-\frac{221}{248}a^{9}+\frac{85}{248}a^{8}+\frac{251}{31}a^{7}-\frac{91}{248}a^{6}-\frac{41}{248}a^{5}+\frac{1137}{124}a^{4}-\frac{1465}{248}a^{3}+\frac{1325}{248}a^{2}-\frac{67}{124}a+\frac{123}{248}$, $a$, $\frac{117}{124}a^{11}-\frac{61}{124}a^{10}-\frac{3}{2}a^{9}+\frac{153}{124}a^{8}+\frac{1431}{124}a^{7}-\frac{159}{31}a^{6}-\frac{579}{124}a^{5}+\frac{2025}{124}a^{4}-\frac{249}{31}a^{3}+\frac{507}{124}a^{2}+\frac{67}{124}a+\frac{63}{62}$, $\frac{61}{248}a^{11}-\frac{47}{124}a^{10}-\frac{151}{248}a^{9}+\frac{221}{248}a^{8}+\frac{427}{124}a^{7}-\frac{1181}{248}a^{6}-\frac{1117}{248}a^{5}+\frac{222}{31}a^{4}-\frac{17}{8}a^{3}-\frac{311}{248}a^{2}+\frac{5}{62}a+\frac{129}{248}$, $\frac{25}{124}a^{11}-\frac{10}{31}a^{10}-\frac{105}{124}a^{9}+\frac{105}{124}a^{8}+\frac{104}{31}a^{7}-\frac{543}{124}a^{6}-\frac{985}{124}a^{5}+\frac{407}{62}a^{4}-\frac{45}{124}a^{3}-\frac{811}{124}a^{2}+\frac{157}{62}a-\frac{349}{124}$, $\frac{25}{62}a^{11}-\frac{157}{124}a^{10}+\frac{13}{124}a^{9}+\frac{137}{62}a^{8}+\frac{387}{124}a^{7}-\frac{1851}{124}a^{6}+\frac{198}{31}a^{5}+\frac{1491}{124}a^{4}-\frac{2981}{124}a^{3}+\frac{456}{31}a^{2}-\frac{479}{124}a-\frac{299}{124}$ | 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: | \( 446.230796412 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 446.230796412 \cdot 1}{2\cdot\sqrt{82608050631424}}\cr\approx \mathstrut & 0.961562881261 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
3.1.76.1, 4.2.27436.1 x2, 6.2.2085136.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 4 sibling: | 4.2.27436.1 |
Degree 6 siblings: | 6.0.109744.1, 6.2.2085136.1 |
Degree 8 sibling: | 8.0.752734096.1 |
Degree 12 sibling: | 12.0.4347792138496.1 |
Minimal sibling: | 4.2.27436.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(19\) | 19.4.3.2 | $x^{4} + 38$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
19.8.6.2 | $x^{8} + 72 x^{7} + 1952 x^{6} + 23760 x^{5} + 112814 x^{4} + 48888 x^{3} + 44288 x^{2} + 435600 x + 1945825$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |