Normalized defining polynomial
\( x^{14} - 3 x^{13} + 10 x^{12} - 21 x^{11} + 39 x^{10} - 60 x^{9} + 84 x^{8} - 93 x^{7} + 84 x^{6} + \cdots + 1 \)
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: | \(-85117432631759703\) \(\medspace = -\,3^{16}\cdot 7^{11}\) | 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: | \(16.19\) | 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: | $3^{4/3}7^{5/6}\approx 21.898281770364438$ | ||
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: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{23}a^{13}-\frac{4}{23}a^{12}-\frac{9}{23}a^{11}+\frac{11}{23}a^{10}+\frac{5}{23}a^{9}+\frac{4}{23}a^{8}+\frac{11}{23}a^{7}+\frac{11}{23}a^{6}+\frac{4}{23}a^{5}+\frac{5}{23}a^{4}+\frac{11}{23}a^{3}-\frac{9}{23}a^{2}-\frac{4}{23}a+\frac{1}{23}$
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: | $a$, $\frac{17}{23}a^{13}-\frac{45}{23}a^{12}+\frac{123}{23}a^{11}-\frac{250}{23}a^{10}+\frac{338}{23}a^{9}-\frac{507}{23}a^{8}+\frac{532}{23}a^{7}-\frac{434}{23}a^{6}-\frac{1}{23}a^{5}+\frac{62}{23}a^{4}-\frac{66}{23}a^{3}-\frac{38}{23}a^{2}-\frac{45}{23}a-\frac{6}{23}$, $\frac{48}{23}a^{13}-\frac{77}{23}a^{12}+\frac{327}{23}a^{11}-\frac{461}{23}a^{10}+\frac{884}{23}a^{9}-\frac{1073}{23}a^{8}+\frac{1471}{23}a^{7}-\frac{967}{23}a^{6}+\frac{698}{23}a^{5}-\frac{197}{23}a^{4}+\frac{206}{23}a^{3}+\frac{51}{23}a^{2}+\frac{15}{23}a+\frac{25}{23}$, $\frac{41}{23}a^{13}-\frac{118}{23}a^{12}+\frac{390}{23}a^{11}-\frac{814}{23}a^{10}+\frac{1470}{23}a^{9}-\frac{2274}{23}a^{8}+\frac{3096}{23}a^{7}-\frac{3367}{23}a^{6}+\frac{2878}{23}a^{5}-\frac{1980}{23}a^{4}+\frac{1095}{23}a^{3}-\frac{461}{23}a^{2}+\frac{112}{23}a-\frac{28}{23}$, $\frac{9}{23}a^{13}-\frac{36}{23}a^{12}+\frac{103}{23}a^{11}-\frac{269}{23}a^{10}+\frac{459}{23}a^{9}-\frac{815}{23}a^{8}+\frac{1111}{23}a^{7}-\frac{1350}{23}a^{6}+\frac{1278}{23}a^{5}-\frac{1036}{23}a^{4}+\frac{582}{23}a^{3}-\frac{265}{23}a^{2}+\frac{79}{23}a-\frac{37}{23}$, $\frac{25}{23}a^{13}-\frac{54}{23}a^{12}+\frac{189}{23}a^{11}-\frac{346}{23}a^{10}+\frac{585}{23}a^{9}-\frac{866}{23}a^{8}+\frac{1103}{23}a^{7}-\frac{990}{23}a^{6}+\frac{744}{23}a^{5}-\frac{404}{23}a^{4}+\frac{160}{23}a^{3}-\frac{18}{23}a^{2}+\frac{15}{23}a+\frac{2}{23}$ | 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: | \( 684.224209859 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 684.224209859 \cdot 1}{2\cdot\sqrt{85117432631759703}}\cr\approx \mathstrut & 0.453333968081 \end{aligned}\]
Galois group
A solvable group of order 42 |
The 7 conjugacy class representatives for $F_7$ |
Character table for $F_7$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 7.1.110270727.1 |
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: | 7.1.110270727.1 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.110270727.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{7}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{7}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.6.8.6 | $x^{6} + 18 x^{5} + 114 x^{4} + 362 x^{3} + 894 x^{2} + 960 x + 557$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ | |
3.6.8.6 | $x^{6} + 18 x^{5} + 114 x^{4} + 362 x^{3} + 894 x^{2} + 960 x + 557$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ | |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.6.5.6 | $x^{6} + 28$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.6.5.6 | $x^{6} + 28$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |