Normalized defining polynomial
\( x^{12} - x^{11} + x^{10} + 4x^{9} + 5x^{8} + 7x^{7} + 15x^{6} + 14x^{5} + 13x^{4} + 11x^{3} + 9x^{2} + 3x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(4077334216081\) \(\medspace = 7^{8}\cdot 29^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.24\) | 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: | $7^{2/3}29^{1/2}\approx 19.705964328162043$ | ||
Ramified primes: | \(7\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7}a^{9}-\frac{3}{7}a^{8}+\frac{1}{7}a^{7}+\frac{3}{7}a^{5}+\frac{2}{7}a^{4}-\frac{2}{7}a^{2}-\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{10}-\frac{1}{7}a^{8}+\frac{3}{7}a^{7}+\frac{3}{7}a^{6}-\frac{3}{7}a^{5}-\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-\frac{1}{7}a^{2}-\frac{3}{7}$, $\frac{1}{7}a^{11}-\frac{3}{7}a^{7}-\frac{3}{7}a^{6}+\frac{2}{7}a^{5}-\frac{1}{7}a^{3}-\frac{2}{7}a^{2}+\frac{2}{7}a-\frac{1}{7}$
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)
| |
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: | $\frac{1}{7}a^{11}+\frac{3}{7}a^{10}-\frac{2}{7}a^{9}+\frac{3}{7}a^{8}+\frac{25}{7}a^{7}+\frac{27}{7}a^{6}+\frac{36}{7}a^{5}+8a^{4}+6a^{3}+\frac{27}{7}a^{2}+\frac{20}{7}a+\frac{6}{7}$, $\frac{5}{7}a^{11}-\frac{9}{7}a^{10}+\frac{10}{7}a^{9}+2a^{8}+\frac{10}{7}a^{7}+3a^{6}+\frac{46}{7}a^{5}+\frac{8}{7}a^{4}+\frac{20}{7}a^{3}+a^{2}+\frac{4}{7}a-\frac{9}{7}$, $\frac{1}{7}a^{11}-\frac{2}{7}a^{10}+\frac{2}{7}a^{8}+\frac{5}{7}a^{7}-\frac{9}{7}a^{6}-\frac{13}{7}a^{5}-\frac{19}{7}a^{4}-\frac{25}{7}a^{3}-5a^{2}-\frac{12}{7}a-\frac{9}{7}$, $\frac{5}{7}a^{11}-\frac{3}{7}a^{10}+\frac{1}{7}a^{9}+3a^{8}+\frac{33}{7}a^{7}+\frac{39}{7}a^{6}+\frac{64}{7}a^{5}+\frac{68}{7}a^{4}+\frac{50}{7}a^{3}+\frac{33}{7}a^{2}+\frac{22}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{10}+\frac{2}{7}a^{9}-a^{8}+\frac{12}{7}a^{7}+\frac{10}{7}a^{6}+\frac{10}{7}a^{5}+\frac{3}{7}a^{4}+\frac{19}{7}a^{3}-\frac{5}{7}a^{2}-\frac{11}{7}a-\frac{5}{7}$ | 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: | \( 20.3909313008 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 20.3909313008 \cdot 1}{2\cdot\sqrt{4077334216081}}\cr\approx \mathstrut & 0.310669143497 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T6):
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
\(\Q(\zeta_{7})^+\), 6.2.69629.1 x2, 6.2.2019241.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.2.69629.1 |
Degree 8 sibling: | 8.0.1698181681.2 |
Degree 12 sibling: | 12.4.3429038075724121.1 |
Minimal sibling: | 6.2.69629.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(29\) | 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |