Normalized defining polynomial
\( x^{14} - 3 x^{12} - x^{11} + 10 x^{10} - x^{9} - 10 x^{8} + 2 x^{7} + 10 x^{6} + 5 x^{5} - 4 x^{4} - 2 x^{3} + 3 x^{2} + 2 x - 1 \)
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: | \(729722900774101\) \(\medspace = 13^{4}\cdot 109^{4}\cdot 181\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.53\) | 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: | $13^{1/2}109^{1/2}181^{1/2}\approx 506.43558326800064$ | ||
Ramified primes: | \(13\), \(109\), \(181\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{181}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{426319}a^{13}+\frac{150375}{426319}a^{12}-\frac{171776}{426319}a^{11}-\frac{147791}{426319}a^{10}-\frac{62145}{426319}a^{9}-\frac{141896}{426319}a^{8}+\frac{81259}{426319}a^{7}+\frac{166949}{426319}a^{6}-\frac{117387}{426319}a^{5}+\frac{94394}{426319}a^{4}+\frac{206641}{426319}a^{3}+\frac{101101}{426319}a^{2}+\frac{101019}{426319}a+\frac{133519}{426319}$
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)
| |
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: | $\frac{46977}{426319}a^{13}+\frac{60545}{426319}a^{12}-\frac{155120}{426319}a^{11}-\frac{172892}{426319}a^{10}+\frac{473166}{426319}a^{9}+\frac{501811}{426319}a^{8}-\frac{808921}{426319}a^{7}-\frac{227470}{426319}a^{6}+\frac{799804}{426319}a^{5}+\frac{1055657}{426319}a^{4}-\frac{335692}{426319}a^{3}-\frac{198302}{426319}a^{2}+\frac{639093}{426319}a+\frac{316935}{426319}$, $\frac{12531}{426319}a^{13}+\frac{19145}{426319}a^{12}-\frac{40425}{426319}a^{11}-\frac{39285}{426319}a^{10}+\frac{145818}{426319}a^{9}+\frac{77773}{426319}a^{8}-\frac{219562}{426319}a^{7}+\frac{90586}{426319}a^{6}+\frac{250372}{426319}a^{5}-\frac{184011}{426319}a^{4}+\frac{383084}{426319}a^{3}-\frac{123437}{426319}a^{2}+\frac{127978}{426319}a+\frac{250833}{426319}$, $a$, $\frac{44371}{426319}a^{13}-\frac{29544}{426319}a^{12}-\frac{141814}{426319}a^{11}+\frac{4397}{426319}a^{10}+\frac{421816}{426319}a^{9}-\frac{188424}{426319}a^{8}-\frac{263013}{426319}a^{7}-\frac{24865}{426319}a^{6}-\frac{239354}{426319}a^{5}+\frac{624637}{426319}a^{4}+\frac{25078}{426319}a^{3}-\frac{628685}{426319}a^{2}-\frac{430236}{426319}a-\frac{183594}{426319}$, $\frac{188630}{426319}a^{13}+\frac{101585}{426319}a^{12}-\frac{583923}{426319}a^{11}-\frac{390601}{426319}a^{10}+\frac{1787469}{426319}a^{9}+\frac{595935}{426319}a^{8}-\frac{2119751}{426319}a^{7}+\frac{257978}{426319}a^{6}+\frac{1578007}{426319}a^{5}+\frac{1606142}{426319}a^{4}+\frac{345660}{426319}a^{3}-\frac{698835}{426319}a^{2}+\frac{459946}{426319}a+\frac{467726}{426319}$, $\frac{190288}{426319}a^{13}+\frac{26720}{426319}a^{12}-\frac{607439}{426319}a^{11}-\frac{294654}{426319}a^{10}+\frac{1920257}{426319}a^{9}+\frac{234136}{426319}a^{8}-\frac{2109133}{426319}a^{7}-\frac{47930}{426319}a^{6}+\frac{1778144}{426319}a^{5}+\frac{1652321}{426319}a^{4}-\frac{1082995}{426319}a^{3}-\frac{616744}{426319}a^{2}+\frac{406081}{426319}a+\frac{156348}{426319}$, $\frac{216005}{426319}a^{13}+\frac{80946}{426319}a^{12}-\frac{653353}{426319}a^{11}-\frac{401916}{426319}a^{10}+\frac{2007223}{426319}a^{9}+\frac{385344}{426319}a^{8}-\frac{2187168}{426319}a^{7}+\frac{347173}{426319}a^{6}+\frac{1701504}{426319}a^{5}+\frac{1296114}{426319}a^{4}-\frac{110095}{426319}a^{3}-\frac{295589}{426319}a^{2}+\frac{323718}{426319}a+\frac{291245}{426319}$ | 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: | \( 44.6471365658 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 44.6471365658 \cdot 1}{2\cdot\sqrt{729722900774101}}\cr\approx \mathstrut & 0.203387298874 \end{aligned}\]
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
7.3.2007889.1 |
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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.14.0.1}{14} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(109\) | $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(181\) | 181.2.1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
181.4.0.1 | $x^{4} + 6 x^{2} + 105 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
181.4.0.1 | $x^{4} + 6 x^{2} + 105 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
181.4.0.1 | $x^{4} + 6 x^{2} + 105 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |