Normalized defining polynomial
\( x^{14} - 3 x^{13} + 5 x^{12} - 2 x^{11} - 5 x^{10} + 12 x^{9} - 3 x^{8} - 3 x^{7} + 11 x^{6} - 4 x^{4} + \cdots - 1 \)
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: | \(1116758251460917\) \(\medspace = 13^{4}\cdot 109^{4}\cdot 277\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.88\) | 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: | $13^{1/2}109^{1/2}277^{1/2}\approx 626.5053870478689$ | ||
Ramified primes: | \(13\), \(109\), \(277\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{277}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{1489}a^{13}+\frac{186}{1489}a^{12}-\frac{577}{1489}a^{11}-\frac{358}{1489}a^{10}-\frac{662}{1489}a^{9}-\frac{30}{1489}a^{8}+\frac{283}{1489}a^{7}-\frac{120}{1489}a^{6}-\frac{334}{1489}a^{5}-\frac{588}{1489}a^{4}+\frac{539}{1489}a^{3}+\frac{618}{1489}a^{2}+\frac{657}{1489}a+\frac{583}{1489}$
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: | $\frac{1178}{1489}a^{13}-\frac{4242}{1489}a^{12}+\frac{8212}{1489}a^{11}-\frac{6293}{1489}a^{10}-\frac{4067}{1489}a^{9}+\frac{18264}{1489}a^{8}-\frac{13563}{1489}a^{7}+\frac{95}{1489}a^{6}+\frac{17512}{1489}a^{5}-\frac{10702}{1489}a^{4}-\frac{861}{1489}a^{3}+\frac{2861}{1489}a^{2}-\frac{4801}{1489}a-\frac{1144}{1489}$, $a$, $\frac{417}{1489}a^{13}-\frac{1355}{1489}a^{12}+\frac{2098}{1489}a^{11}-\frac{386}{1489}a^{10}-\frac{3567}{1489}a^{9}+\frac{6847}{1489}a^{8}-\frac{2598}{1489}a^{7}-\frac{2392}{1489}a^{6}+\frac{5155}{1489}a^{5}-\frac{2489}{1489}a^{4}-\frac{76}{1489}a^{3}-\frac{1380}{1489}a^{2}-\frac{1496}{1489}a+\frac{404}{1489}$, $\frac{898}{1489}a^{13}-\frac{4207}{1489}a^{12}+\frac{10449}{1489}a^{11}-\frac{14750}{1489}a^{10}+\frac{10058}{1489}a^{9}+\frac{5818}{1489}a^{8}-\frac{16864}{1489}a^{7}+\frac{14338}{1489}a^{6}+\frac{846}{1489}a^{5}-\frac{8363}{1489}a^{4}+\frac{3075}{1489}a^{3}+\frac{1056}{1489}a^{2}-\frac{2636}{1489}a-\frac{594}{1489}$, $\frac{58}{1489}a^{13}+\frac{365}{1489}a^{12}-\frac{2197}{1489}a^{11}+\frac{6038}{1489}a^{10}-\frac{8616}{1489}a^{9}+\frac{5705}{1489}a^{8}+\frac{4502}{1489}a^{7}-\frac{9938}{1489}a^{6}+\frac{7430}{1489}a^{5}+\frac{3121}{1489}a^{4}-\frac{4474}{1489}a^{3}+\frac{1597}{1489}a^{2}+\frac{881}{1489}a-\frac{433}{1489}$, $\frac{1085}{1489}a^{13}-\frac{3672}{1489}a^{12}+\frac{6780}{1489}a^{11}-\frac{4268}{1489}a^{10}-\frac{5039}{1489}a^{9}+\frac{16587}{1489}a^{8}-\frac{10102}{1489}a^{7}-\frac{657}{1489}a^{6}+\frac{14327}{1489}a^{5}-\frac{5155}{1489}a^{4}-\frac{3340}{1489}a^{3}+\frac{1969}{1489}a^{2}-\frac{3364}{1489}a-\frac{3248}{1489}$, $\frac{2387}{1489}a^{13}-\frac{8674}{1489}a^{12}+\frac{17894}{1489}a^{11}-\frac{17728}{1489}a^{10}+\frac{2613}{1489}a^{9}+\frac{23686}{1489}a^{8}-\frac{21331}{1489}a^{7}+\frac{9871}{1489}a^{6}+\frac{17225}{1489}a^{5}-\frac{8363}{1489}a^{4}-\frac{2881}{1489}a^{3}-\frac{433}{1489}a^{2}-\frac{7103}{1489}a-\frac{5061}{1489}$ | 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: | \( 56.9116802426 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 56.9116802426 \cdot 1}{2\cdot\sqrt{1116758251460917}}\cr\approx \mathstrut & 0.209570924170 \end{aligned}\]
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
7.3.2007889.2 |
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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | 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 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(109\) | $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
109.8.4.1 | $x^{8} + 28776 x^{7} + 310522274 x^{6} + 1489272514288 x^{5} + 2678510521605233 x^{4} + 178743008720712 x^{3} + 29612720181709536 x^{2} + 263388846138180416 x + 20054316486246464$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(277\) | $\Q_{277}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{277}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |