Normalized defining polynomial
\( x^{15} - 2484 x^{13} - 39744 x^{12} + 1586292 x^{11} + 55530624 x^{10} + 421080120 x^{9} - 7864615296 x^{8} - 219580868736 x^{7} - 2464952159232 x^{6} - 15966575400960 x^{5} - 64404675624960 x^{4} - 164230567526400 x^{3} - 257781340569600 x^{2} - 227693297664000 x - 86740303872000 \)
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: | \(626870103658230384189023990781533773422403883667456000000=2^{18}\cdot 3^{24}\cdot 5^{6}\cdot 17^{4}\cdot 37^{2}\cdot 521\cdot 811^{4}\cdot 4585573^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $6116.16$ | 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: | $2, 3, 5, 17, 37, 521, 811, 4585573$ | 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 \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{40} a^{6} + \frac{1}{20} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{160} a^{7} - \frac{1}{40} a^{5} - \frac{1}{20} a^{4} - \frac{3}{40} a^{3} - \frac{1}{5} a^{2} - \frac{1}{4} a$, $\frac{1}{3200} a^{8} - \frac{9}{800} a^{6} + \frac{1}{50} a^{5} - \frac{99}{800} a^{4} + \frac{6}{25} a^{3} + \frac{23}{400} a^{2} + \frac{1}{5}$, $\frac{1}{102400} a^{9} - \frac{69}{25600} a^{7} - \frac{9}{1600} a^{6} - \frac{579}{25600} a^{5} - \frac{39}{800} a^{4} + \frac{1303}{12800} a^{3} + \frac{33}{160} a^{2} + \frac{41}{160} a - \frac{1}{4}$, $\frac{1}{6963200} a^{10} + \frac{123}{1740800} a^{8} + \frac{191}{108800} a^{7} - \frac{2371}{1740800} a^{6} - \frac{19}{10880} a^{5} + \frac{5671}{51200} a^{4} + \frac{389}{3200} a^{3} + \frac{237}{3200} a^{2} + \frac{3}{16} a - \frac{2}{5}$, $\frac{1}{1114112000} a^{11} - \frac{1}{27852800} a^{10} + \frac{599}{278528000} a^{9} + \frac{1943}{34816000} a^{8} + \frac{309773}{278528000} a^{7} + \frac{11331}{2048000} a^{6} - \frac{43961}{5570560} a^{5} + \frac{28583}{1024000} a^{4} + \frac{61987}{256000} a^{3} - \frac{267}{2000} a^{2} - \frac{2429}{6400} a + \frac{137}{800}$, $\frac{1}{22282240000} a^{12} + \frac{199}{5570560000} a^{10} + \frac{429}{348160000} a^{9} - \frac{461107}{5570560000} a^{8} + \frac{499293}{174080000} a^{7} + \frac{304067}{557056000} a^{6} + \frac{2294833}{174080000} a^{5} - \frac{198743}{5120000} a^{4} + \frac{156493}{640000} a^{3} + \frac{4387}{128000} a^{2} - \frac{1617}{4000} a + \frac{133}{400}$, $\frac{1}{44564480000000} a^{13} - \frac{3}{278528000000} a^{12} + \frac{67}{655360000000} a^{11} + \frac{27909}{696320000000} a^{10} + \frac{2848413}{11141120000000} a^{9} - \frac{5772241}{174080000000} a^{8} - \frac{3119941413}{1114112000000} a^{7} - \frac{524655837}{348160000000} a^{6} + \frac{8462053623}{348160000000} a^{5} - \frac{90048349}{2560000000} a^{4} - \frac{41918883}{256000000} a^{3} - \frac{842019}{8000000} a^{2} + \frac{881233}{3200000} a - \frac{193321}{400000}$, $\frac{1}{45634027520000000} a^{14} - \frac{43}{5704253440000000} a^{13} - \frac{233181}{11408506880000000} a^{12} + \frac{410181}{1426063360000000} a^{11} + \frac{134850397}{11408506880000000} a^{10} + \frac{3973529093}{1426063360000000} a^{9} + \frac{107407592103}{5704253440000000} a^{8} - \frac{1131421813779}{713031680000000} a^{7} - \frac{336149837709}{356515840000000} a^{6} + \frac{239357587227}{11141120000000} a^{5} + \frac{3487909853}{1310720000000} a^{4} + \frac{5995148013}{32768000000} a^{3} - \frac{2147622503}{16384000000} a^{2} + \frac{1062309}{40960000} a + \frac{5079793}{51200000}$
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: | \( 104236366388000000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 5184000 |
| The 133 conjugacy class representatives for [S(5)^3]3=S(5)wr3 are not computed |
| Character table for [S(5)^3]3=S(5)wr3 is not computed |
Intermediate fields
| \(\Q(\zeta_{9})^+\) |
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 siblings: | 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 | R | R | ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ | $15$ | R | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ | ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ | $15$ | ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.6.9.1 | $x^{6} + 4 x^{4} + 4 x^{2} - 8$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
| 2.6.9.1 | $x^{6} + 4 x^{4} + 4 x^{2} - 8$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
| 3 | Data not computed | ||||||
| $5$ | 5.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 5.6.3.2 | $x^{6} - 25 x^{2} + 250$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.2 | $x^{6} - 25 x^{2} + 250$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $17$ | 17.5.4.1 | $x^{5} - 17$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 17.5.0.1 | $x^{5} - x + 6$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 17.5.0.1 | $x^{5} - x + 6$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| $37$ | $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.2.2 | $x^{4} - 37 x^{2} + 6845$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 37.5.0.1 | $x^{5} - x + 13$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 521 | Data not computed | ||||||
| 811 | Data not computed | ||||||
| 4585573 | Data not computed | ||||||