Normalized defining polynomial
\( x^{15} + 12 x^{13} - 8 x^{12} + 9 x^{11} - 12 x^{10} - 185 x^{9} + 378 x^{8} - 495 x^{7} + 704 x^{6} - 405 x^{5} - 522 x^{4} + 1032 x^{3} - 720 x^{2} + 240 x - 32 \)
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[7, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2262670157726939535197184=2^{10}\cdot 3^{12}\cdot 401^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $42.04$ | 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: | $2, 3, 401$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{8} a^{11} - \frac{1}{2} a^{9} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{12} + \frac{1}{32} a^{11} - \frac{7}{16} a^{10} + \frac{9}{64} a^{8} - \frac{13}{32} a^{7} + \frac{7}{64} a^{6} + \frac{3}{8} a^{5} + \frac{29}{64} a^{4} + \frac{1}{32} a^{3} - \frac{13}{64} a^{2} - \frac{1}{16} a - \frac{5}{16}$, $\frac{1}{1536} a^{13} + \frac{1}{384} a^{12} - \frac{11}{192} a^{11} + \frac{89}{192} a^{10} - \frac{253}{512} a^{9} + \frac{47}{192} a^{8} + \frac{275}{1536} a^{7} - \frac{111}{256} a^{6} + \frac{269}{1536} a^{5} + \frac{175}{384} a^{4} + \frac{125}{512} a^{3} - \frac{175}{768} a^{2} + \frac{169}{384} a - \frac{53}{192}$, $\frac{1}{12288} a^{14} - \frac{1}{6144} a^{13} + \frac{5}{768} a^{12} + \frac{11}{1536} a^{11} - \frac{1933}{4096} a^{10} - \frac{2143}{6144} a^{9} - \frac{3325}{12288} a^{8} - \frac{481}{1024} a^{7} + \frac{4073}{12288} a^{6} - \frac{457}{6144} a^{5} - \frac{1979}{4096} a^{4} - \frac{661}{1536} a^{3} - \frac{157}{1536} a^{2} + \frac{31}{96} a - \frac{27}{256}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $10$ | 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: | \( 7043298.15376 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 38880 |
| The 48 conjugacy class representatives for [1/2.S(3)^5]D(5) |
| Character table for [1/2.S(3)^5]D(5) is not computed |
Intermediate fields
| 5.5.160801.1 |
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 | R | $15$ | $15$ | $15$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $15$ | ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | $15$ | $15$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 2.10.10.1 | $x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$ | $2$ | $5$ | $10$ | $C_2^4 : C_5$ | $[2, 2, 2, 2]^{5}$ | |
| $3$ | 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.3.4.3 | $x^{3} - 3 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.6.6.1 | $x^{6} + 3 x^{5} - 2$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
| 401 | Data not computed | ||||||