Normalized defining polynomial
\( x^{14} - 14 x^{11} - 14 x^{10} - 28 x^{9} + 35 x^{8} + 83 x^{7} + 245 x^{6} + 294 x^{5} + 399 x^{4} + \cdots + 57 \)
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: | \(-72680419155717387\) \(\medspace = -\,3^{7}\cdot 7^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.01\) | 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: | $3^{1/2}7^{26/21}\approx 19.26959054025096$ | ||
Ramified primes: | \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{139}a^{13}+\frac{60}{139}a^{12}-\frac{14}{139}a^{11}-\frac{20}{139}a^{10}+\frac{37}{139}a^{9}-\frac{32}{139}a^{8}+\frac{61}{139}a^{7}-\frac{10}{139}a^{6}+\frac{62}{139}a^{5}-\frac{17}{139}a^{4}-\frac{65}{139}a^{3}+\frac{15}{139}a^{2}+\frac{47}{139}a+\frac{6}{139}$
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: | \( a^{7} - 7 a^{4} - 7 a^{3} - 14 a^{2} - 7 a - 7 \) (order $6$) | 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: | $\frac{91}{139}a^{13}-\frac{100}{139}a^{12}-\frac{23}{139}a^{11}-\frac{1125}{139}a^{10}+\frac{31}{139}a^{9}-\frac{827}{139}a^{8}+\frac{4439}{139}a^{7}+\frac{3955}{139}a^{6}+\frac{10785}{139}a^{5}+\frac{6376}{139}a^{4}+\frac{9097}{139}a^{3}+\frac{2338}{139}a^{2}+\frac{2609}{139}a-\frac{149}{139}$, $\frac{3}{139}a^{13}+\frac{41}{139}a^{12}-\frac{42}{139}a^{11}-\frac{60}{139}a^{10}-\frac{584}{139}a^{9}-\frac{96}{139}a^{8}-\frac{234}{139}a^{7}+\frac{2472}{139}a^{6}+\frac{2827}{139}a^{5}+\frac{6343}{139}a^{4}+\frac{4670}{139}a^{3}+\frac{5883}{139}a^{2}+\frac{2365}{139}a+\frac{1964}{139}$, $\frac{50}{139}a^{13}-\frac{58}{139}a^{12}-\frac{5}{139}a^{11}-\frac{583}{139}a^{10}+\frac{43}{139}a^{9}-\frac{627}{139}a^{8}+\frac{2077}{139}a^{7}+\frac{1863}{139}a^{6}+\frac{6436}{139}a^{5}+\frac{4710}{139}a^{4}+\frac{7036}{139}a^{3}+\frac{2974}{139}a^{2}+\frac{2767}{139}a+\frac{161}{139}$, $\frac{20}{139}a^{13}-\frac{51}{139}a^{12}-\frac{2}{139}a^{11}-\frac{261}{139}a^{10}+\frac{462}{139}a^{9}+\frac{194}{139}a^{8}+\frac{1776}{139}a^{7}-\frac{617}{139}a^{6}-\frac{428}{139}a^{5}-\frac{5622}{139}a^{4}-\frac{4636}{139}a^{3}-\frac{7067}{139}a^{2}-\frac{3091}{139}a-\frac{2521}{139}$, $\frac{9}{139}a^{13}-\frac{16}{139}a^{12}+\frac{13}{139}a^{11}-\frac{41}{139}a^{10}-\frac{84}{139}a^{9}-\frac{149}{139}a^{8}-\frac{146}{139}a^{7}+\frac{605}{139}a^{6}+\frac{975}{139}a^{5}+\frac{2349}{139}a^{4}+\frac{2056}{139}a^{3}+\frac{2359}{139}a^{2}+\frac{979}{139}a+\frac{610}{139}$, $\frac{46}{139}a^{13}-\frac{159}{139}a^{12}+\frac{190}{139}a^{11}-\frac{642}{139}a^{10}+\frac{1146}{139}a^{9}-\frac{1055}{139}a^{8}+\frac{3362}{139}a^{7}-\frac{2267}{139}a^{6}+\frac{3269}{139}a^{5}-\frac{6481}{139}a^{4}-\frac{627}{139}a^{3}-\frac{8206}{139}a^{2}-\frac{1591}{139}a-\frac{3616}{139}$ | 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: | \( 2216.78301748 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 2216.78301748 \cdot 1}{6\cdot\sqrt{72680419155717387}}\cr\approx \mathstrut & 0.529812624063 \end{aligned}\]
Galois group
A solvable group of order 42 |
The 7 conjugacy class representatives for $F_7$ |
Character table for $F_7$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 7.1.155649627.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 7 sibling: | 7.1.155649627.1 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.155649627.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }^{7}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.7.8.3 | $x^{7} + 28 x^{2} + 7$ | $7$ | $1$ | $8$ | $C_7:C_3$ | $[4/3]_{3}$ |
7.7.8.3 | $x^{7} + 28 x^{2} + 7$ | $7$ | $1$ | $8$ | $C_7:C_3$ | $[4/3]_{3}$ |