Normalized defining polynomial
\( x^{14} - 2 x^{13} - x^{12} + 3 x^{11} - 3 x^{10} + 4 x^{9} + x^{8} + 5 x^{7} + x^{6} + 4 x^{5} + \cdots + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(842608211391089\) \(\medspace = 11\cdot 13^{4}\cdot 19\cdot 109^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.64\) | 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: | $11^{1/2}13^{1/2}19^{1/2}109^{1/2}\approx 544.1994119805717$ | ||
Ramified primes: | \(11\), \(13\), \(19\), \(109\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{209}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{113}a^{12}+\frac{15}{113}a^{11}+\frac{27}{113}a^{10}-\frac{5}{113}a^{9}-\frac{2}{113}a^{8}-\frac{25}{113}a^{7}+\frac{30}{113}a^{6}-\frac{25}{113}a^{5}-\frac{2}{113}a^{4}-\frac{5}{113}a^{3}+\frac{27}{113}a^{2}+\frac{15}{113}a+\frac{1}{113}$, $\frac{1}{113}a^{13}+\frac{28}{113}a^{11}+\frac{42}{113}a^{10}-\frac{40}{113}a^{9}+\frac{5}{113}a^{8}-\frac{47}{113}a^{7}-\frac{23}{113}a^{6}+\frac{34}{113}a^{5}+\frac{25}{113}a^{4}-\frac{11}{113}a^{3}-\frac{51}{113}a^{2}+\frac{2}{113}a-\frac{15}{113}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{24}{113}a^{13}-\frac{34}{113}a^{12}-\frac{64}{113}a^{11}+\frac{90}{113}a^{10}+\frac{1}{113}a^{9}-\frac{38}{113}a^{8}+\frac{61}{113}a^{7}+\frac{123}{113}a^{6}+\frac{84}{113}a^{5}+\frac{216}{113}a^{4}+\frac{132}{113}a^{3}+\frac{118}{113}a^{2}+\frac{216}{113}a-\frac{55}{113}$, $a$, $\frac{6}{113}a^{13}+\frac{13}{113}a^{12}+\frac{24}{113}a^{11}-\frac{75}{113}a^{10}-\frac{192}{113}a^{9}+\frac{4}{113}a^{8}+\frac{184}{113}a^{7}+\frac{365}{113}a^{6}+\frac{444}{113}a^{5}+\frac{463}{113}a^{4}+\frac{434}{113}a^{3}+\frac{271}{113}a^{2}-\frac{19}{113}a-\frac{190}{113}$, $\frac{37}{113}a^{13}-\frac{8}{113}a^{12}-\frac{101}{113}a^{11}-\frac{131}{113}a^{10}+\frac{29}{113}a^{9}+\frac{314}{113}a^{8}+\frac{156}{113}a^{7}+\frac{378}{113}a^{6}+\frac{215}{113}a^{5}+\frac{376}{113}a^{4}-\frac{28}{113}a^{3}+\frac{44}{113}a^{2}-\frac{159}{113}a+\frac{115}{113}$, $\frac{63}{113}a^{13}-\frac{61}{113}a^{12}-\frac{168}{113}a^{11}+\frac{95}{113}a^{10}-\frac{68}{113}a^{9}+\frac{98}{113}a^{8}+\frac{372}{113}a^{7}+\frac{450}{113}a^{6}+\frac{390}{113}a^{5}+\frac{454}{113}a^{4}+\frac{177}{113}a^{3}-\frac{1}{113}a^{2}+\frac{115}{113}a-\frac{102}{113}$, $\frac{18}{113}a^{13}-\frac{78}{113}a^{12}+\frac{12}{113}a^{11}+\frac{232}{113}a^{10}-\frac{104}{113}a^{9}-\frac{93}{113}a^{8}-\frac{139}{113}a^{7}-\frac{42}{113}a^{6}-\frac{37}{113}a^{5}-\frac{72}{113}a^{4}+\frac{79}{113}a^{3}+\frac{253}{113}a^{2}+\frac{222}{113}a-\frac{122}{113}$, $\frac{8}{113}a^{13}-\frac{53}{113}a^{12}-\frac{6}{113}a^{11}+\frac{148}{113}a^{10}+\frac{58}{113}a^{9}+\frac{33}{113}a^{8}-\frac{181}{113}a^{7}-\frac{305}{113}a^{6}-\frac{324}{113}a^{5}-\frac{259}{113}a^{4}-\frac{388}{113}a^{3}-\frac{31}{113}a^{2}+\frac{12}{113}a+\frac{53}{113}$ | 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: | \( 49.4412723787 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 49.4412723787 \cdot 1}{2\cdot\sqrt{842608211391089}}\cr\approx \mathstrut & 0.209597453128 \end{aligned}\]
Galois group
$C_2^7.\GL(3,2)$ (as 14T51):
A non-solvable group of order 21504 |
The 48 conjugacy class representatives for $C_2^7.\GL(3,2)$ |
Character table for $C_2^7.\GL(3,2)$ |
Intermediate fields
7.3.2007889.1 |
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 | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | R | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | R | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(19\) | 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
19.8.0.1 | $x^{8} + x^{4} + 12 x^{3} + 10 x^{2} + 3 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
\(109\) | 109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
109.4.2.1 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
109.4.0.1 | $x^{4} + 11 x^{2} + 98 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
109.4.2.1 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |