Normalized defining polynomial
\( x^{14} - 3 x^{13} + 2 x^{12} + 3 x^{11} - 5 x^{10} - x^{9} + x^{8} + 2 x^{7} + 11 x^{6} - 13 x^{5} + \cdots + 1 \)
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)
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Discriminant: | \(-118236562059083\) \(\medspace = -\,13\cdot 71^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.12\) | 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: | $13^{1/2}71^{3/4}\approx 88.18920705302763$ | ||
Ramified primes: | \(13\), \(71\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-923}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13823}a^{13}+\frac{6906}{13823}a^{12}-\frac{3440}{13823}a^{11}-\frac{5220}{13823}a^{10}-\frac{778}{13823}a^{9}+\frac{1944}{13823}a^{8}-\frac{4859}{13823}a^{7}+\frac{5238}{13823}a^{6}+\frac{739}{13823}a^{5}+\frac{5051}{13823}a^{4}-\frac{5710}{13823}a^{3}+\frac{444}{13823}a^{2}-\frac{1106}{13823}a+\frac{2765}{13823}$
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: | $\frac{1893}{13823}a^{13}-\frac{17323}{13823}a^{12}+\frac{26359}{13823}a^{11}+\frac{1985}{13823}a^{10}-\frac{48985}{13823}a^{9}+\frac{30720}{13823}a^{8}+\frac{35677}{13823}a^{7}+\frac{4443}{13823}a^{6}+\frac{30450}{13823}a^{5}-\frac{100734}{13823}a^{4}+\frac{83494}{13823}a^{3}-\frac{44180}{13823}a^{2}+\frac{21261}{13823}a-\frac{4772}{13823}$, $\frac{4547}{13823}a^{13}-\frac{18097}{13823}a^{12}+\frac{19779}{13823}a^{11}+\frac{12574}{13823}a^{10}-\frac{40347}{13823}a^{9}+\frac{6471}{13823}a^{8}+\frac{22927}{13823}a^{7}+\frac{13980}{13823}a^{6}+\frac{28890}{13823}a^{5}-\frac{117513}{13823}a^{4}+\frac{65339}{13823}a^{3}-\frac{26936}{13823}a^{2}+\frac{44059}{13823}a-\frac{6475}{13823}$, $\frac{909}{13823}a^{13}+\frac{1912}{13823}a^{12}-\frac{2962}{13823}a^{11}-\frac{3691}{13823}a^{10}+\frac{11594}{13823}a^{9}-\frac{2248}{13823}a^{8}-\frac{21117}{13823}a^{7}-\frac{7593}{13823}a^{6}-\frac{5576}{13823}a^{5}+\frac{15946}{13823}a^{4}+\frac{20881}{13823}a^{3}-\frac{11094}{13823}a^{2}+\frac{17548}{13823}a-\frac{2401}{13823}$, $\frac{9289}{13823}a^{13}-\frac{30355}{13823}a^{12}+\frac{18439}{13823}a^{11}+\frac{43973}{13823}a^{10}-\frac{66528}{13823}a^{9}-\frac{22668}{13823}a^{8}+\frac{52136}{13823}a^{7}+\frac{26468}{13823}a^{6}+\frac{77478}{13823}a^{5}-\frac{148576}{13823}a^{4}+\frac{26307}{13823}a^{3}-\frac{8761}{13823}a^{2}+\frac{10678}{13823}a+\frac{14774}{13823}$, $\frac{4742}{13823}a^{13}-\frac{12258}{13823}a^{12}-\frac{1340}{13823}a^{11}+\frac{31399}{13823}a^{10}-\frac{26181}{13823}a^{9}-\frac{29139}{13823}a^{8}+\frac{29209}{13823}a^{7}+\frac{12488}{13823}a^{6}+\frac{48588}{13823}a^{5}-\frac{31063}{13823}a^{4}-\frac{39032}{13823}a^{3}+\frac{18175}{13823}a^{2}-\frac{33381}{13823}a+\frac{21249}{13823}$, $\frac{3966}{13823}a^{13}-\frac{7990}{13823}a^{12}+\frac{261}{13823}a^{11}+\frac{18157}{13823}a^{10}-\frac{16842}{13823}a^{9}-\frac{17153}{13823}a^{8}+\frac{12291}{13823}a^{7}-\frac{2061}{13823}a^{6}+\frac{14221}{13823}a^{5}-\frac{24907}{13823}a^{4}+\frac{10037}{13823}a^{3}+\frac{33029}{13823}a^{2}-\frac{4505}{13823}a+\frac{4351}{13823}$ | 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: | \( 12.9746502268 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 12.9746502268 \cdot 1}{2\cdot\sqrt{118236562059083}}\cr\approx \mathstrut & 0.230647549616 \end{aligned}\]
Galois group
$C_2\wr D_7$ (as 14T38):
A solvable group of order 1792 |
The 40 conjugacy class representatives for $C_2\wr D_7$ |
Character table for $C_2\wr D_7$ is not computed |
Intermediate fields
7.1.357911.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 14 siblings: | data not computed |
Degree 28 siblings: | data not computed |
Degree 32 sibling: | 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.14.0.1}{14} }$ | ${\href{/padicField/23.2.0.1}{2} }^{7}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(71\) | $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
71.4.3.1 | $x^{4} + 71$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |