Normalized defining polynomial
\( x^{14} - 3 x^{13} - 5 x^{12} + 18 x^{11} + 14 x^{10} - 52 x^{9} - 27 x^{8} + 91 x^{7} + 31 x^{6} + \cdots + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-417195690175859375\) \(\medspace = -\,5^{7}\cdot 11\cdot 2939\cdot 5351\cdot 30869\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.14\) | 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: | $5^{1/2}11^{1/2}2939^{1/2}5351^{1/2}30869^{1/2}\approx 5167254.993829412$ | ||
Ramified primes: | \(5\), \(11\), \(2939\), \(5351\), \(30869\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-26700524171255}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2122}a^{13}-\frac{173}{1061}a^{12}+\frac{451}{1061}a^{11}+\frac{222}{1061}a^{10}+\frac{253}{1061}a^{9}+\frac{197}{1061}a^{8}+\frac{639}{2122}a^{7}-\frac{260}{1061}a^{6}-\frac{459}{1061}a^{5}+\frac{366}{1061}a^{4}-\frac{691}{2122}a^{3}-\frac{611}{2122}a^{2}+\frac{553}{2122}a+\frac{116}{1061}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $10$ | 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: | $\frac{835}{1061}a^{13}-\frac{3501}{1061}a^{12}-\frac{140}{1061}a^{11}+\frac{15305}{1061}a^{10}-\frac{5073}{1061}a^{9}-\frac{38116}{1061}a^{8}+\frac{16858}{1061}a^{7}+\frac{57043}{1061}a^{6}-\frac{28074}{1061}a^{5}-\frac{38112}{1061}a^{4}+\frac{17175}{1061}a^{3}+\frac{8644}{1061}a^{2}-\frac{2963}{1061}a-\frac{443}{1061}$, $\frac{1161}{1061}a^{13}-\frac{8723}{2122}a^{12}-\frac{2107}{1061}a^{11}+\frac{21058}{1061}a^{10}-\frac{328}{1061}a^{9}-\frac{53968}{1061}a^{8}+\frac{9789}{1061}a^{7}+\frac{166555}{2122}a^{6}-\frac{24957}{1061}a^{5}-\frac{59425}{1061}a^{4}+\frac{26390}{1061}a^{3}+\frac{27401}{2122}a^{2}-\frac{17781}{2122}a+\frac{777}{2122}$, $\frac{777}{2122}a^{13}-\frac{4653}{2122}a^{12}+\frac{2419}{1061}a^{11}+\frac{9100}{1061}a^{10}-\frac{15619}{1061}a^{9}-\frac{19874}{1061}a^{8}+\frac{86957}{2122}a^{7}+\frac{51129}{2122}a^{6}-\frac{71234}{1061}a^{5}-\frac{8454}{1061}a^{4}+\frac{110303}{2122}a^{3}-\frac{10850}{1061}a^{2}-\frac{14866}{1061}a+\frac{11565}{2122}$, $\frac{1161}{1061}a^{13}-\frac{8723}{2122}a^{12}-\frac{2107}{1061}a^{11}+\frac{21058}{1061}a^{10}-\frac{328}{1061}a^{9}-\frac{53968}{1061}a^{8}+\frac{9789}{1061}a^{7}+\frac{166555}{2122}a^{6}-\frac{24957}{1061}a^{5}-\frac{59425}{1061}a^{4}+\frac{26390}{1061}a^{3}+\frac{27401}{2122}a^{2}-\frac{15659}{2122}a+\frac{777}{2122}$, $\frac{1095}{1061}a^{13}-\frac{9735}{2122}a^{12}+\frac{960}{1061}a^{11}+\frac{20401}{1061}a^{10}-\frac{12504}{1061}a^{9}-\frac{49203}{1061}a^{8}+\frac{38702}{1061}a^{7}+\frac{141827}{2122}a^{6}-\frac{65164}{1061}a^{5}-\frac{43016}{1061}a^{4}+\frac{48654}{1061}a^{3}+\frac{12563}{2122}a^{2}-\frac{24995}{2122}a+\frac{1983}{2122}$, $\frac{47}{1061}a^{13}-\frac{1755}{2122}a^{12}+\frac{3137}{1061}a^{11}-\frac{352}{1061}a^{10}-\frac{12292}{1061}a^{9}+\frac{6847}{1061}a^{8}+\frac{28972}{1061}a^{7}-\frac{43575}{2122}a^{6}-\frac{39963}{1061}a^{5}+\frac{35465}{1061}a^{4}+\frac{20573}{1061}a^{3}-\frac{39397}{2122}a^{2}-\frac{4251}{2122}a+\frac{5893}{2122}$, $\frac{1309}{2122}a^{13}-\frac{2586}{1061}a^{12}+\frac{443}{1061}a^{11}+\frac{7311}{1061}a^{10}-\frac{1977}{1061}a^{9}-\frac{12682}{1061}a^{8}+\frac{8871}{2122}a^{7}+\frac{6607}{1061}a^{6}-\frac{2427}{1061}a^{5}+\frac{17559}{1061}a^{4}-\frac{19645}{2122}a^{3}-\frac{16781}{2122}a^{2}+\frac{13007}{2122}a-\frac{940}{1061}$, $\frac{31}{2122}a^{13}+\frac{945}{2122}a^{12}-\frac{1934}{1061}a^{11}-\frac{545}{1061}a^{10}+\frac{7843}{1061}a^{9}-\frac{259}{1061}a^{8}-\frac{35363}{2122}a^{7}+\frac{6161}{2122}a^{6}+\frac{22906}{1061}a^{5}-\frac{7752}{1061}a^{4}-\frac{17177}{2122}a^{3}+\frac{8036}{1061}a^{2}+\frac{614}{1061}a-\frac{2357}{2122}$, $\frac{1547}{2122}a^{13}-\frac{7945}{2122}a^{12}+\frac{3803}{1061}a^{11}+\frac{10280}{1061}a^{10}-\frac{16033}{1061}a^{9}-\frac{17785}{1061}a^{8}+\frac{76073}{2122}a^{7}+\frac{19957}{2122}a^{6}-\frac{46948}{1061}a^{5}+\frac{20848}{1061}a^{4}+\frac{19609}{2122}a^{3}-\frac{11604}{1061}a^{2}+\frac{4937}{1061}a-\frac{2897}{2122}$, $\frac{1659}{2122}a^{13}-\frac{6379}{2122}a^{12}-\frac{857}{1061}a^{11}+\frac{13924}{1061}a^{10}-\frac{2551}{1061}a^{9}-\frac{32856}{1061}a^{8}+\frac{24565}{2122}a^{7}+\frac{91159}{2122}a^{6}-\frac{26208}{1061}a^{5}-\frac{21979}{1061}a^{4}+\frac{48317}{2122}a^{3}-\frac{1258}{1061}a^{2}-\frac{5474}{1061}a+\frac{3989}{2122}$ | 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: | \( 4127.10213237 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{3}\cdot 4127.10213237 \cdot 1}{2\cdot\sqrt{417195690175859375}}\cr\approx \mathstrut & 0.202873295713 \end{aligned}\]
Galois group
$S_7\wr C_2$ (as 14T61):
A non-solvable group of order 50803200 |
The 135 conjugacy class representatives for $S_7\wr C_2$ |
Character table for $S_7\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 28 siblings: | data not computed |
Degree 42 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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.14.0.1}{14} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
\(11\) | 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.5.0.1 | $x^{5} + 10 x^{2} + 9$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
11.7.0.1 | $x^{7} + 4 x + 9$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(2939\) | $\Q_{2939}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2939}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(5351\) | $\Q_{5351}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(30869\) | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |