Normalized defining polynomial
\( x^{15} - 846 x^{13} - 6768 x^{12} + 141712 x^{11} + 2470432 x^{10} + 17068680 x^{9} + 97426368 x^{8} + 617654944 x^{7} + 3048750464 x^{6} + 9692440960 x^{5} + 19512852480 x^{4} + 24878681600 x^{3} + 19525171200 x^{2} + 8623104000 x + 1642496000 \)
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[11, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(4997116069370523770922817361794182094079365120000000=2^{18}\cdot 5^{7}\cdot 7^{2}\cdot 11^{5}\cdot 79^{5}\cdot 401^{4}\cdot 19713263^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $2796.28$ | 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: | $2, 5, 7, 11, 79, 401, 19713263$ | 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 \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} - \frac{1}{40} a^{5} + \frac{1}{20} a^{4} - \frac{1}{2} a$, $\frac{1}{800} a^{8} + \frac{3}{400} a^{6} + \frac{1}{100} a^{5} + \frac{2}{25} a^{4} - \frac{1}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} - \frac{7}{1600} a^{7} - \frac{1}{100} a^{6} + \frac{1}{50} a^{5} + \frac{3}{25} a^{4} - \frac{3}{400} a^{3} + \frac{1}{5} a^{2} + \frac{9}{20} a$, $\frac{1}{32000} a^{10} - \frac{1}{16000} a^{9} + \frac{9}{16000} a^{8} - \frac{41}{8000} a^{7} - \frac{1}{200} a^{6} - \frac{7}{500} a^{5} - \frac{3}{4000} a^{4} - \frac{449}{2000} a^{3} + \frac{201}{1000} a^{2} - \frac{9}{100} a + \frac{6}{25}$, $\frac{1}{128000} a^{11} + \frac{7}{64000} a^{9} - \frac{3}{8000} a^{8} - \frac{61}{16000} a^{7} - \frac{9}{4000} a^{6} + \frac{33}{3200} a^{5} + \frac{217}{2000} a^{4} - \frac{111}{500} a^{3} - \frac{87}{1000} a^{2} - \frac{97}{200} a + \frac{1}{50}$, $\frac{1}{1280000} a^{12} + \frac{7}{640000} a^{10} - \frac{3}{80000} a^{9} - \frac{61}{160000} a^{8} + \frac{191}{40000} a^{7} - \frac{287}{32000} a^{6} - \frac{83}{20000} a^{5} + \frac{589}{5000} a^{4} - \frac{387}{10000} a^{3} + \frac{263}{2000} a^{2} + \frac{101}{500} a + \frac{1}{5}$, $\frac{1}{5120000000} a^{13} - \frac{31}{128000000} a^{12} + \frac{5457}{2560000000} a^{11} - \frac{1829}{160000000} a^{10} + \frac{1167}{320000000} a^{9} + \frac{25021}{160000000} a^{8} - \frac{680227}{128000000} a^{7} + \frac{19223}{20000000} a^{6} + \frac{1606657}{160000000} a^{5} - \frac{3859947}{40000000} a^{4} - \frac{1738549}{8000000} a^{3} - \frac{116539}{1000000} a^{2} + \frac{39299}{400000} a - \frac{12563}{100000}$, $\frac{1}{3276800000000} a^{14} - \frac{3}{32768000000} a^{13} - \frac{37343}{1638400000000} a^{12} + \frac{1315079}{409600000000} a^{11} - \frac{1812353}{204800000000} a^{10} + \frac{3913511}{102400000000} a^{9} - \frac{3993687}{16384000000} a^{8} - \frac{266472941}{102400000000} a^{7} - \frac{1103366383}{102400000000} a^{6} + \frac{753941}{200000000} a^{5} + \frac{43003083}{1024000000} a^{4} - \frac{257464093}{1280000000} a^{3} + \frac{12651887}{51200000} a^{2} - \frac{2886649}{32000000} a - \frac{1454061}{3200000}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $12$ | 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: | \( 2643567297610000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 5184000 |
| The 79 conjugacy class representatives for [1/2.S(5)^3]S(3) are not computed |
| Character table for [1/2.S(5)^3]S(3) is not computed |
Intermediate fields
| 3.3.4345.1 |
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 | ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | R | R | R | ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | $15$ | ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ | $15$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.6.9.6 | $x^{6} + 4 x^{2} + 8$ | $2$ | $3$ | $9$ | $A_4\times C_2$ | $[2, 2, 3]^{3}$ | |
| 2.6.9.6 | $x^{6} + 4 x^{2} + 8$ | $2$ | $3$ | $9$ | $A_4\times C_2$ | $[2, 2, 3]^{3}$ | |
| $5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.8.0.1 | $x^{8} - x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $11$ | $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 11.2.1.2 | $x^{2} + 33$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 79 | Data not computed | ||||||
| 401 | Data not computed | ||||||
| 19713263 | Data not computed | ||||||