Normalized defining polynomial
\( x^{14} - 2x^{13} + 3x^{11} - 7x^{10} + x^{9} + 14x^{8} - 2x^{7} - 13x^{6} + 2x^{5} + 4x^{4} - 4x^{3} + 3x + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-498897726810659\) \(\medspace = -\,11^{3}\cdot 71^{2}\cdot 8623^{2}\) | 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: | \(11.22\) | 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: | $11^{1/2}71^{1/2}8623^{1/2}\approx 2595.103658815963$ | ||
Ramified primes: | \(11\), \(71\), \(8623\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{31}a^{13}+\frac{3}{31}a^{12}+\frac{15}{31}a^{11}-\frac{15}{31}a^{10}+\frac{11}{31}a^{9}-\frac{6}{31}a^{8}+\frac{15}{31}a^{7}+\frac{11}{31}a^{6}+\frac{11}{31}a^{5}-\frac{5}{31}a^{4}+\frac{10}{31}a^{3}+\frac{15}{31}a^{2}+\frac{13}{31}a+\frac{6}{31}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $\frac{54}{31}a^{13}-\frac{117}{31}a^{12}+\frac{35}{31}a^{11}+\frac{151}{31}a^{10}-\frac{429}{31}a^{9}+\frac{172}{31}a^{8}+\frac{655}{31}a^{7}-\frac{336}{31}a^{6}-\frac{491}{31}a^{5}+\frac{288}{31}a^{4}+\frac{75}{31}a^{3}-\frac{244}{31}a^{2}+\frac{113}{31}a+\frac{107}{31}$, $\frac{58}{31}a^{13}-\frac{136}{31}a^{12}+\frac{64}{31}a^{11}+\frac{122}{31}a^{10}-\frac{447}{31}a^{9}+\frac{241}{31}a^{8}+\frac{622}{31}a^{7}-\frac{323}{31}a^{6}-\frac{478}{31}a^{5}+\frac{330}{31}a^{4}+\frac{22}{31}a^{3}-\frac{246}{31}a^{2}+\frac{103}{31}a+\frac{100}{31}$, $\frac{25}{31}a^{13}-\frac{80}{31}a^{12}+\frac{34}{31}a^{11}+\frac{90}{31}a^{10}-\frac{252}{31}a^{9}+\frac{191}{31}a^{8}+\frac{437}{31}a^{7}-\frac{345}{31}a^{6}-\frac{407}{31}a^{5}+\frac{309}{31}a^{4}+\frac{64}{31}a^{3}-\frac{214}{31}a^{2}+\frac{108}{31}a+\frac{88}{31}$, $\frac{46}{31}a^{13}-\frac{110}{31}a^{12}+\frac{101}{31}a^{11}+\frac{23}{31}a^{10}-\frac{331}{31}a^{9}+\frac{282}{31}a^{8}+\frac{194}{31}a^{7}-\frac{238}{31}a^{6}-\frac{52}{31}a^{5}+\frac{235}{31}a^{4}-\frac{129}{31}a^{3}-\frac{54}{31}a^{2}+\frac{71}{31}a-\frac{34}{31}$, $\frac{88}{31}a^{13}-\frac{232}{31}a^{12}+\frac{142}{31}a^{11}+\frac{168}{31}a^{10}-\frac{706}{31}a^{9}+\frac{526}{31}a^{8}+\frac{917}{31}a^{7}-\frac{675}{31}a^{6}-\frac{737}{31}a^{5}+\frac{552}{31}a^{4}-\frac{19}{31}a^{3}-\frac{354}{31}a^{2}+\frac{183}{31}a+\frac{156}{31}$, $\frac{30}{31}a^{13}-\frac{34}{31}a^{12}+\frac{16}{31}a^{11}+\frac{15}{31}a^{10}-\frac{135}{31}a^{9}-\frac{56}{31}a^{8}+\frac{109}{31}a^{7}+\frac{144}{31}a^{6}-\frac{11}{31}a^{5}+\frac{36}{31}a^{4}-\frac{41}{31}a^{3}-\frac{46}{31}a^{2}-\frac{75}{31}a-\frac{6}{31}$, $\frac{53}{31}a^{13}-\frac{151}{31}a^{12}+\frac{82}{31}a^{11}+\frac{135}{31}a^{10}-\frac{471}{31}a^{9}+\frac{364}{31}a^{8}+\frac{671}{31}a^{7}-\frac{533}{31}a^{6}-\frac{564}{31}a^{5}+\frac{417}{31}a^{4}+\frac{65}{31}a^{3}-\frac{259}{31}a^{2}+\frac{100}{31}a+\frac{101}{31}$, $\frac{1}{31}a^{13}+\frac{3}{31}a^{12}+\frac{15}{31}a^{11}-\frac{46}{31}a^{10}+\frac{42}{31}a^{9}-\frac{6}{31}a^{8}-\frac{140}{31}a^{7}+\frac{135}{31}a^{6}+\frac{135}{31}a^{5}-\frac{129}{31}a^{4}-\frac{83}{31}a^{3}+\frac{108}{31}a^{2}-\frac{18}{31}a-\frac{56}{31}$ | 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: | \( 47.3515916708 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{5}\cdot 47.3515916708 \cdot 1}{2\cdot\sqrt{498897726810659}}\cr\approx \mathstrut & 0.166080310507 \end{aligned}\]
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
7.3.612233.1 |
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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.6.3.1 | $x^{6} + 242 x^{2} - 11979$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
11.8.0.1 | $x^{8} + 7 x^{4} + 7 x^{3} + x^{2} + 7 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
\(71\) | 71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.5.0.1 | $x^{5} + 18 x + 64$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
71.5.0.1 | $x^{5} + 18 x + 64$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(8623\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |