Normalized defining polynomial
\( x^{14} + 7 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-65712362363534280139543\) \(\medspace = -\,7^{27}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(42.64\) | 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: | $7^{83/42}\approx 46.781559808244616$ | ||
Ramified primes: | \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{6}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{1}{2}a^{12}+a^{11}+a^{10}+\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{3}{2}a^{7}-2a^{6}-\frac{3}{2}a^{5}+2a^{3}+\frac{7}{2}a^{2}+\frac{7}{2}a+\frac{5}{2}$, $\frac{1}{2}a^{13}+\frac{1}{2}a^{12}+a^{11}+\frac{1}{2}a^{10}+2a^{9}+a^{8}+\frac{5}{2}a^{7}+\frac{1}{2}a^{6}+\frac{5}{2}a^{5}+\frac{5}{2}a^{3}-\frac{5}{2}$, $a^{13}-\frac{3}{2}a^{11}-\frac{3}{2}a^{10}+\frac{1}{2}a^{9}+\frac{5}{2}a^{8}+2a^{7}-2a^{6}-3a^{5}-\frac{7}{2}a^{4}+\frac{11}{2}a^{3}+\frac{7}{2}a^{2}+\frac{7}{2}a-8$, $\frac{1}{2}a^{12}+\frac{3}{2}a^{11}+\frac{5}{2}a^{10}+\frac{7}{2}a^{9}+5a^{8}+6a^{7}+8a^{6}+\frac{17}{2}a^{5}+\frac{21}{2}a^{4}+\frac{19}{2}a^{3}+\frac{21}{2}a^{2}+7a+6$, $a^{13}+a^{12}+\frac{3}{2}a^{11}+\frac{1}{2}a^{10}-2a^{9}-\frac{3}{2}a^{8}-\frac{5}{2}a^{7}-3a^{6}+2a^{5}+\frac{7}{2}a^{4}+\frac{9}{2}a^{3}+7a^{2}+\frac{7}{2}a-\frac{5}{2}$, $\frac{9}{2}a^{13}+\frac{11}{2}a^{12}+7a^{11}+10a^{10}+\frac{23}{2}a^{9}+10a^{8}+\frac{19}{2}a^{7}+\frac{25}{2}a^{6}+\frac{27}{2}a^{5}+7a^{4}-a^{3}-\frac{7}{2}a^{2}-7a-\frac{37}{2}$ (assuming GRH) | 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: | \( 3075737.72709 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 3075737.72709 \cdot 1}{2\cdot\sqrt{65712362363534280139543}}\cr\approx \mathstrut & 2.31928886215 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 42 |
The 7 conjugacy class representatives for $F_7$ |
Character table for $F_7$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 7.1.96889010407.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 42 |
Degree 7 sibling: | 7.1.96889010407.1 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.96889010407.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.1.0.1}{1} }^{14}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.14.27.54 | $x^{14} + 49 x^{2} + 301$ | $14$ | $1$ | $27$ | $F_7$ | $[13/6]_{6}$ |