Normalized defining polynomial
\( x^{15} + 9 x^{13} - 15 x^{12} - 12 x^{11} - 90 x^{10} + 104 x^{9} - 213 x^{8} + 159 x^{7} - 165 x^{6} + 918 x^{5} - 1329 x^{4} + 1957 x^{3} - 546 x^{2} + 636 x + 178 \)
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: | \(343441853189647952875776=2^{8}\cdot 3^{24}\cdot 41^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $37.07$ | 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: | $2, 3, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{17489863795625986419824} a^{14} - \frac{1631117237496702955489}{17489863795625986419824} a^{13} + \frac{1923185176473904107745}{8744931897812993209912} a^{12} - \frac{686787536396887813361}{17489863795625986419824} a^{11} - \frac{1734196399122812990739}{17489863795625986419824} a^{10} + \frac{609727648807484044461}{1345374138125075878448} a^{9} + \frac{45577561676313201139}{1345374138125075878448} a^{8} - \frac{353632872875686075059}{4372465948906496604956} a^{7} - \frac{6666593006061341096501}{17489863795625986419824} a^{6} + \frac{454216569380597728883}{1093116487226624151239} a^{5} - \frac{3324176588694533967701}{8744931897812993209912} a^{4} - \frac{140901901965974764421}{296438369417389600336} a^{3} + \frac{101255475441942456061}{4372465948906496604956} a^{2} + \frac{1780187665925893560021}{8744931897812993209912} a + \frac{713724919651201855105}{8744931897812993209912}$
Class group and class number
Trivial group, which has order $1$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 5595418.09537 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 233280 |
| The 48 conjugacy class representatives for [1/2.S(3)^5]A(5) |
| Character table for [1/2.S(3)^5]A(5) is not computed |
Intermediate fields
| 5.1.1225449.1 |
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 | ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ | $15$ | $15$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ | $15$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ | $15$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ | $15$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 2.6.6.1 | $x^{6} + x^{2} - 1$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ | |
| $3$ | 3.6.9.16 | $x^{6} + 3 x^{4} + 6 x^{3} + 3$ | $6$ | $1$ | $9$ | $S_3^2$ | $[3/2, 2]_{2}^{2}$ |
| 3.9.15.3 | $x^{9} + 9 x^{6} + 72 x^{3} + 27$ | $3$ | $3$ | $15$ | $S_3\times C_3$ | $[5/2]_{2}^{3}$ | |
| 41 | Data not computed | ||||||