Normalized defining polynomial
\( x^{15} - 20 x^{13} - 39 x^{12} + 160 x^{11} + 624 x^{10} - 173 x^{9} - 3744 x^{8} - 4324 x^{7} + 7911 x^{6} + 21392 x^{5} + 6600 x^{4} - 26128 x^{3} - 33168 x^{2} - 15040 x - 2304 \)
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[11, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(126641586691841137969338115072=2^{10}\cdot 881^{6}\cdot 3769\cdot 70177\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $87.13$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 881, 3769, 70177$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{4} a^{8} + \frac{1}{4} a^{5} - \frac{1}{3} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{1776} a^{12} - \frac{1}{111} a^{10} - \frac{7}{1776} a^{9} + \frac{2}{37} a^{8} + \frac{7}{148} a^{7} + \frac{193}{592} a^{6} - \frac{7}{37} a^{5} + \frac{85}{222} a^{4} + \frac{77}{592} a^{3} - \frac{53}{111} a^{2} + \frac{217}{444} a - \frac{17}{37}$, $\frac{1}{3552} a^{13} - \frac{1}{222} a^{11} - \frac{7}{3552} a^{10} + \frac{1}{37} a^{9} + \frac{7}{296} a^{8} + \frac{193}{1184} a^{7} - \frac{7}{74} a^{6} + \frac{85}{444} a^{5} + \frac{77}{1184} a^{4} - \frac{53}{222} a^{3} + \frac{217}{888} a^{2} + \frac{10}{37} a$, $\frac{1}{7104} a^{14} - \frac{7}{7104} a^{11} - \frac{5}{222} a^{10} - \frac{7}{1776} a^{9} + \frac{113}{2368} a^{8} + \frac{21}{148} a^{7} + \frac{355}{888} a^{6} - \frac{531}{2368} a^{5} + \frac{61}{148} a^{4} - \frac{635}{1776} a^{3} + \frac{211}{444} a^{2} - \frac{5}{111} a + \frac{6}{37}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $12$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 44248875484.8 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 466560 |
| The 72 conjugacy class representatives for [S(3)^5]A(5)=S(3)wrA(5) are not computed |
| Character table for [S(3)^5]A(5)=S(3)wrA(5) is not computed |
Intermediate fields
| 5.5.3104644.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ | $15$ | ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ | ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ | ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | $15$ | ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.4.4.2 | $x^{4} - x^{2} + 5$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 881 | Data not computed | ||||||
| 3769 | Data not computed | ||||||
| 70177 | Data not computed | ||||||