Normalized defining polynomial
\( x^{14} - 4 x^{13} + 3 x^{12} + 4 x^{11} - 2 x^{10} - 2 x^{9} - 6 x^{8} + 2 x^{7} + x^{6} + 7 x^{5} + \cdots + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1616678912764721\) \(\medspace = 13^{4}\cdot 109^{4}\cdot 401\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.20\) | 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: | $13^{1/2}109^{1/2}401^{1/2}\approx 753.8016980612341$ | ||
Ramified primes: | \(13\), \(109\), \(401\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{401}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13}a^{13}+\frac{5}{13}a^{12}-\frac{4}{13}a^{11}-\frac{6}{13}a^{10}-\frac{4}{13}a^{9}+\frac{1}{13}a^{8}+\frac{3}{13}a^{7}+\frac{3}{13}a^{6}+\frac{2}{13}a^{5}-\frac{1}{13}a^{4}+\frac{3}{13}a^{2}-\frac{5}{13}a-\frac{3}{13}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{2}{13}a^{13}+\frac{10}{13}a^{12}-\frac{21}{13}a^{11}-\frac{25}{13}a^{10}+\frac{5}{13}a^{9}+\frac{41}{13}a^{8}+\frac{58}{13}a^{7}+\frac{6}{13}a^{6}-\frac{48}{13}a^{5}-\frac{119}{13}a^{4}-7a^{3}+\frac{6}{13}a^{2}+\frac{55}{13}a+\frac{33}{13}$, $\frac{25}{13}a^{13}-\frac{44}{13}a^{12}-\frac{61}{13}a^{11}+\frac{71}{13}a^{10}+\frac{95}{13}a^{9}+\frac{38}{13}a^{8}-\frac{68}{13}a^{7}-\frac{146}{13}a^{6}-\frac{119}{13}a^{5}-\frac{51}{13}a^{4}+17a^{3}+\frac{179}{13}a^{2}-\frac{99}{13}a-\frac{49}{13}$, $\frac{7}{13}a^{13}-\frac{17}{13}a^{12}+\frac{11}{13}a^{11}-\frac{16}{13}a^{10}+\frac{11}{13}a^{9}+\frac{20}{13}a^{8}+\frac{8}{13}a^{7}+\frac{8}{13}a^{6}-\frac{51}{13}a^{5}-\frac{20}{13}a^{4}-3a^{3}-\frac{18}{13}a^{2}+\frac{30}{13}a+\frac{18}{13}$, $a$, $\frac{6}{13}a^{13}-\frac{9}{13}a^{12}+\frac{2}{13}a^{11}-\frac{23}{13}a^{10}+\frac{2}{13}a^{9}+\frac{45}{13}a^{8}+\frac{18}{13}a^{7}+\frac{18}{13}a^{6}-\frac{66}{13}a^{5}-\frac{58}{13}a^{4}-5a^{3}-\frac{21}{13}a^{2}+\frac{61}{13}a+\frac{8}{13}$, $a^{13}-2a^{12}-2a^{11}+3a^{10}+4a^{9}+a^{8}-4a^{7}-6a^{6}-4a^{5}+10a^{3}+7a^{2}-5a-3$, $\frac{14}{13}a^{13}-\frac{34}{13}a^{12}+\frac{22}{13}a^{11}-\frac{32}{13}a^{10}+\frac{22}{13}a^{9}+\frac{40}{13}a^{8}+\frac{16}{13}a^{7}+\frac{16}{13}a^{6}-\frac{102}{13}a^{5}-\frac{53}{13}a^{4}-4a^{3}-\frac{36}{13}a^{2}+\frac{60}{13}a+\frac{23}{13}$, $\frac{45}{13}a^{13}-\frac{126}{13}a^{12}+\frac{15}{13}a^{11}+\frac{120}{13}a^{10}+\frac{54}{13}a^{9}+\frac{32}{13}a^{8}-\frac{190}{13}a^{7}-\frac{86}{13}a^{6}-\frac{144}{13}a^{5}+\frac{85}{13}a^{4}+28a^{3}-\frac{34}{13}a^{2}-\frac{121}{13}a-\frac{5}{13}$, $\frac{17}{13}a^{13}-\frac{45}{13}a^{12}+\frac{23}{13}a^{11}-\frac{11}{13}a^{10}+\frac{23}{13}a^{9}+\frac{43}{13}a^{8}-\frac{27}{13}a^{7}+\frac{12}{13}a^{6}-\frac{109}{13}a^{5}-\frac{17}{13}a^{4}-\frac{27}{13}a^{2}+\frac{58}{13}a+\frac{14}{13}$ | 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: | \( 124.900795328 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{4}\cdot 124.900795328 \cdot 1}{2\cdot\sqrt{1616678912764721}}\cr\approx \mathstrut & 0.154925450889 \end{aligned}\]
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 $C_2^7.\GL(3,2)$ is not computed |
Intermediate fields
7.3.2007889.2 |
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 | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.14.0.1}{14} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(109\) | $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
109.8.4.1 | $x^{8} + 28776 x^{7} + 310522274 x^{6} + 1489272514288 x^{5} + 2678510521605233 x^{4} + 178743008720712 x^{3} + 29612720181709536 x^{2} + 263388846138180416 x + 20054316486246464$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(401\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |