Normalized defining polynomial
\( x^{14} - 3 x^{13} + 6 x^{12} - 11 x^{11} + 16 x^{10} - 21 x^{9} + 25 x^{8} - 25 x^{7} + 25 x^{6} - 21 x^{5} + 16 x^{4} - 11 x^{3} + 6 x^{2} - 3 x + 1 \)
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: | \(-9866941650479\) \(\medspace = -\,167^{2}\cdot 191\cdot 1361^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(8.48\) | 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: | $167^{1/2}191^{1/2}1361^{1/2}\approx 6588.764451701093$ | ||
Ramified primes: | \(167\), \(191\), \(1361\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-191}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$
Monogenic: | Yes | |
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: | $a$, $a^{13}-2a^{12}+4a^{11}-7a^{10}+8a^{9}-11a^{8}+12a^{7}-10a^{6}+12a^{5}-7a^{4}+7a^{3}-4a^{2}+a-1$, $a^{13}-a^{12}+a^{11}-2a^{10}-a^{7}+4a^{6}+4a^{4}-a^{3}+a^{2}-a-1$, $a^{13}-a^{12}+a^{11}-3a^{10}+2a^{9}-3a^{8}+4a^{7}-a^{6}+5a^{5}-a^{4}+a^{3}-2a^{2}-1$, $a^{13}-2a^{12}+3a^{11}-6a^{10}+8a^{9}-10a^{8}+12a^{7}-11a^{6}+12a^{5}-9a^{4}+7a^{3}-5a^{2}+3a-2$, $a^{12}-2a^{11}+4a^{10}-8a^{9}+10a^{8}-13a^{7}+15a^{6}-13a^{5}+14a^{4}-9a^{3}+7a^{2}-5a+1$ | 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: | \( 2.68866215877 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 2.68866215877 \cdot 1}{2\cdot\sqrt{9866941650479}}\cr\approx \mathstrut & 0.165452768164 \end{aligned}\]
Galois group
$C_2^7.S_7$ (as 14T57):
A non-solvable group of order 645120 |
The 110 conjugacy class representatives for $C_2^7.S_7$ are not computed |
Character table for $C_2^7.S_7$ is not computed |
Intermediate fields
7.1.227287.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 14 sibling: | data not computed |
Degree 28 siblings: | data not computed |
Degree 42 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(167\) | 167.3.0.1 | $x^{3} + 7 x + 162$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
167.3.0.1 | $x^{3} + 7 x + 162$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
167.4.0.1 | $x^{4} + 3 x^{2} + 120 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(191\) | 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
191.2.0.1 | $x^{2} + 190 x + 19$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
191.10.0.1 | $x^{10} + 113 x^{5} + 47 x^{4} + 173 x^{3} + 74 x^{2} + 156 x + 19$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(1361\) | $\Q_{1361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |