Normalized defining polynomial
\( x^{20} - 10 x^{19} + 41 x^{18} - 84 x^{17} + 71 x^{16} + 44 x^{15} - 141 x^{14} + 13 x^{13} + 280 x^{12} - 289 x^{11} - 199 x^{10} + 626 x^{9} - 393 x^{8} - 235 x^{7} + 468 x^{6} - 132 x^{5} - 134 x^{4} + 75 x^{3} + 6 x^{2} - 8 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(823067302269314181883621609=11^{18}\cdot 23^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $22.17$ | 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: | $11, 23$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} - \frac{8}{23} a^{15} - \frac{11}{23} a^{14} + \frac{10}{23} a^{13} + \frac{10}{23} a^{12} - \frac{2}{23} a^{11} + \frac{11}{23} a^{10} + \frac{5}{23} a^{9} - \frac{7}{23} a^{8} - \frac{3}{23} a^{7} + \frac{10}{23} a^{6} + \frac{7}{23} a^{5} + \frac{6}{23} a^{4} + \frac{6}{23} a^{3} + \frac{11}{23} a^{2} - \frac{7}{23}$, $\frac{1}{23} a^{17} - \frac{6}{23} a^{15} - \frac{9}{23} a^{14} - \frac{2}{23} a^{13} + \frac{9}{23} a^{12} - \frac{5}{23} a^{11} + \frac{1}{23} a^{10} + \frac{10}{23} a^{9} + \frac{10}{23} a^{8} + \frac{9}{23} a^{7} - \frac{5}{23} a^{6} - \frac{7}{23} a^{5} + \frac{8}{23} a^{4} - \frac{10}{23} a^{3} - \frac{4}{23} a^{2} - \frac{7}{23} a - \frac{10}{23}$, $\frac{1}{529} a^{18} - \frac{9}{529} a^{17} + \frac{3}{529} a^{16} + \frac{180}{529} a^{15} + \frac{164}{529} a^{14} - \frac{251}{529} a^{13} + \frac{142}{529} a^{12} + \frac{28}{529} a^{11} - \frac{107}{529} a^{10} - \frac{150}{529} a^{9} + \frac{109}{529} a^{8} - \frac{205}{529} a^{7} - \frac{56}{529} a^{6} - \frac{165}{529} a^{5} - \frac{189}{529} a^{4} + \frac{232}{529} a^{3} - \frac{240}{529} a^{2} - \frac{16}{529} a + \frac{73}{529}$, $\frac{1}{12167} a^{19} + \frac{2}{12167} a^{18} + \frac{65}{12167} a^{17} + \frac{167}{12167} a^{16} + \frac{488}{12167} a^{15} - \frac{2564}{12167} a^{14} + \frac{3476}{12167} a^{13} + \frac{2579}{12167} a^{12} - \frac{1570}{12167} a^{11} - \frac{1143}{12167} a^{10} + \frac{200}{529} a^{9} - \frac{6067}{12167} a^{8} + \frac{5624}{12167} a^{7} - \frac{5749}{12167} a^{6} + \frac{3424}{12167} a^{5} - \frac{2951}{12167} a^{4} - \frac{632}{12167} a^{3} + \frac{426}{12167} a^{2} + \frac{1944}{12167} a + \frac{2689}{12167}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 1532218.96034 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 5120 |
| The 44 conjugacy class representatives for t20n314 |
| Character table for t20n314 is not computed |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.8.1247354328539.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 11 | Data not computed | ||||||
| $23$ | $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.3.2 | $x^{4} - 23$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |