Normalized defining polynomial
\( x^{15} - 2 x^{14} - 41 x^{13} + 125 x^{12} + 437 x^{11} - 2057 x^{10} + 153 x^{9} + 10144 x^{8} - 16303 x^{7} + 721 x^{6} + 23221 x^{5} - 28390 x^{4} + 16191 x^{3} - 4896 x^{2} + 736 x - 41 \)
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[15, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(28424095136369691319629376=2^{6}\cdot 7^{10}\cdot 11^{6}\cdot 31^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $49.76$ | 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: | $2, 7, 11, 31$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{132441017874101} a^{14} - \frac{39194675530110}{132441017874101} a^{13} + \frac{10426717420919}{132441017874101} a^{12} - \frac{23863081481156}{132441017874101} a^{11} - \frac{29458270830169}{132441017874101} a^{10} + \frac{28869756498471}{132441017874101} a^{9} - \frac{47432036721214}{132441017874101} a^{8} - \frac{38049751215541}{132441017874101} a^{7} + \frac{53419629597697}{132441017874101} a^{6} - \frac{45520827028165}{132441017874101} a^{5} + \frac{8500748405465}{132441017874101} a^{4} + \frac{42370190172775}{132441017874101} a^{3} + \frac{58353564638802}{132441017874101} a^{2} + \frac{53152691149944}{132441017874101} a - \frac{51884770063908}{132441017874101}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 94463462.964 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times A_5$ (as 15T16):
| A non-solvable group of order 180 |
| The 15 conjugacy class representatives for $\GL(2,4)$ |
| Character table for $\GL(2,4)$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 5.5.22791076.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 15 siblings: | data not computed |
| Degree 18 sibling: | data not computed |
| Degree 30 sibling: | 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 | $15$ | $15$ | R | R | ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ | $15$ | $15$ | $15$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ | R | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.0.1 | $x^{6} - x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 2.9.6.1 | $x^{9} - 4 x^{3} + 8$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
| $7$ | 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| $11$ | 11.6.0.1 | $x^{6} + x^{2} - 2 x + 8$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 11.9.6.1 | $x^{9} - 121 x^{3} + 3993$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
| $31$ | 31.3.0.1 | $x^{3} - x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 31.3.0.1 | $x^{3} - x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 31.9.6.1 | $x^{9} + 837 x^{6} + 232562 x^{3} + 21717639$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |