Normalized defining polynomial
\( x^{15} - 2754 x^{13} - 11016 x^{12} + 1610262 x^{11} + 12964716 x^{10} - 84517920 x^{9} - 1422699984 x^{8} - 7303718236 x^{7} - 19757797408 x^{6} - 31834343560 x^{5} - 32087024640 x^{4} - 20455309400 x^{3} - 8026820400 x^{2} - 1772484000 x - 168808000 \)
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: | \(263067183205872529212832748534985625689305855623168000000=2^{22}\cdot 5^{6}\cdot 23^{2}\cdot 37^{5}\cdot 491^{2}\cdot 21101^{4}\cdot 1513121^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $5772.15$ | 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: | $2, 5, 23, 37, 491, 21101, 1513121$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{10} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{10} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{20} a^{7} - \frac{1}{10} a^{4} + \frac{3}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{100} a^{8} + \frac{1}{25} a^{6} - \frac{1}{25} a^{5} - \frac{4}{25} a^{4} - \frac{7}{25} a^{3} - \frac{9}{25} a^{2} + \frac{2}{5}$, $\frac{1}{100} a^{9} - \frac{1}{100} a^{7} - \frac{1}{25} a^{6} + \frac{1}{25} a^{5} - \frac{9}{50} a^{4} - \frac{13}{50} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{32600} a^{10} - \frac{31}{16300} a^{8} - \frac{287}{16300} a^{7} + \frac{7}{326} a^{6} - \frac{91}{8150} a^{5} + \frac{161}{1630} a^{4} + \frac{2367}{8150} a^{3} + \frac{1409}{8150} a^{2} + \frac{59}{163} a - \frac{51}{815}$, $\frac{1}{163000} a^{11} - \frac{357}{81500} a^{9} + \frac{101}{40750} a^{8} - \frac{1769}{81500} a^{7} - \frac{453}{20375} a^{6} + \frac{161}{8150} a^{5} - \frac{5131}{40750} a^{4} + \frac{7958}{20375} a^{3} - \frac{6186}{20375} a^{2} - \frac{703}{4075} a + \frac{6}{25}$, $\frac{1}{815000} a^{12} + \frac{1}{815000} a^{10} + \frac{1017}{407500} a^{9} - \frac{1929}{407500} a^{8} + \frac{8143}{407500} a^{7} - \frac{1383}{40750} a^{6} + \frac{5599}{203750} a^{5} - \frac{36059}{203750} a^{4} - \frac{93407}{203750} a^{3} + \frac{243}{40750} a^{2} + \frac{1633}{20375} a + \frac{694}{4075}$, $\frac{1}{1630000000} a^{13} - \frac{31}{81500000} a^{12} + \frac{93}{815000000} a^{11} - \frac{1321}{101875000} a^{10} - \frac{2218489}{815000000} a^{9} - \frac{217591}{407500000} a^{8} - \frac{886609}{40750000} a^{7} - \frac{580823}{12734375} a^{6} - \frac{16159839}{407500000} a^{5} + \frac{2232847}{101875000} a^{4} - \frac{8041151}{40750000} a^{3} + \frac{3172839}{10187500} a^{2} + \frac{857401}{8150000} a - \frac{1217337}{4075000}$, $\frac{1}{2790560000000} a^{14} + \frac{53}{1395280000000} a^{13} + \frac{374033}{1395280000000} a^{12} + \frac{76539}{27905600000} a^{11} - \frac{4116857}{1395280000000} a^{10} + \frac{137606951}{348820000000} a^{9} + \frac{552182961}{174410000000} a^{8} + \frac{3214933581}{174410000000} a^{7} + \frac{743586009}{27905600000} a^{6} - \frac{7119904863}{348820000000} a^{5} + \frac{2871352089}{348820000000} a^{4} - \frac{1146789387}{6976400000} a^{3} - \frac{34702184539}{69764000000} a^{2} + \frac{1115197813}{3488200000} a + \frac{724820669}{3488200000}$
Class group and class number
$C_{3}$, which has order $3$ (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: | \( 515423356395000000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 1296000 |
| The 65 conjugacy class representatives for [A(5)^3]S(3)=A(5)wrS(3) are not computed |
| Character table for [A(5)^3]S(3)=A(5)wrS(3) is not computed |
Intermediate fields
| 3.3.148.1 |
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 | $15$ | R | ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | $15$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.12.20.69 | $x^{12} + 2 x^{11} + 2 x^{10} + 2 x^{9} + 2 x^{6} + 2 x^{4} + 2$ | $12$ | $1$ | $20$ | 12T206 | $[4/3, 4/3, 4/3, 4/3, 2, 2]_{3}^{6}$ | |
| $5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $23$ | $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.10.0.1 | $x^{10} - x + 7$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| $37$ | 37.5.0.1 | $x^{5} - x + 13$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 37.10.5.1 | $x^{10} - 2738 x^{6} + 1874161 x^{2} - 11719128733$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 491 | Data not computed | ||||||
| 21101 | Data not computed | ||||||
| 1513121 | Data not computed | ||||||