Normalized defining polynomial
\( x^{15} - 10 x^{12} - 20 x^{11} - 76 x^{10} - 200 x^{9} - 450 x^{8} - 655 x^{7} + 590 x^{6} + 3016 x^{5} + 4700 x^{4} + 2425 x^{3} - 2080 x^{2} - 220 x - 8 \)
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[3, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2259801992000000000000000=2^{18}\cdot 5^{15}\cdot 7^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $42.03$ | 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: | $2, 5, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{74} a^{13} - \frac{11}{37} a^{12} + \frac{15}{37} a^{11} - \frac{3}{37} a^{10} + \frac{17}{37} a^{9} - \frac{12}{37} a^{8} - \frac{6}{37} a^{7} - \frac{10}{37} a^{6} - \frac{21}{74} a^{5} - \frac{3}{37} a^{4} + \frac{14}{37} a^{3} - \frac{1}{74} a + \frac{7}{37}$, $\frac{1}{41764753226357636285564} a^{14} - \frac{65442586021971659129}{20882376613178818142782} a^{13} + \frac{5070378918920009204366}{10441188306589409071391} a^{12} + \frac{6390859762760751700411}{20882376613178818142782} a^{11} + \frac{2212175260472572198321}{10441188306589409071391} a^{10} + \frac{459195988151430895873}{10441188306589409071391} a^{9} - \frac{4663443110830038268118}{10441188306589409071391} a^{8} + \frac{415634190400482349635}{1606336662552216780214} a^{7} + \frac{4546704853159379922669}{41764753226357636285564} a^{6} + \frac{5085927477401363152759}{10441188306589409071391} a^{5} + \frac{177754890142928748662}{10441188306589409071391} a^{4} + \frac{1952485006718114660508}{10441188306589409071391} a^{3} - \frac{5689550045756063739299}{41764753226357636285564} a^{2} - \frac{8016315623449063365213}{20882376613178818142782} a - \frac{1480429524239101471732}{10441188306589409071391}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $8$ | 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: | \( 2093567.33677 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 1296000 |
| The 53 conjugacy class representatives for [A(5)^3:2]3 are not computed |
| Character table for [A(5)^3:2]3 is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 45 sibling: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ | R | R | $15$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ | $15$ | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | $15$ | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.12.18.64 | $x^{12} + 14 x^{11} + 12 x^{10} + 4 x^{9} + 10 x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 16 x - 8$ | $4$ | $3$ | $18$ | 12T51 | $[2, 2, 2, 2]^{6}$ | |
| 5 | Data not computed | ||||||
| $7$ | 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |