Normalized defining polynomial
\( x^{20} - 3 x^{19} - 37 x^{18} + 156 x^{17} + 267 x^{16} - 1891 x^{15} + 140 x^{14} + 9731 x^{13} - 7675 x^{12} - 24030 x^{11} + 29458 x^{10} + 27931 x^{9} - 46585 x^{8} - 13731 x^{7} + 34996 x^{6} + 1527 x^{5} - 12441 x^{4} + 450 x^{3} + 1829 x^{2} - 17 x - 79 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[20, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2706114834006068266421672175378489=3^{6}\cdot 61^{4}\cdot 401^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $46.95$ | 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: | $3, 61, 401$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{16490760440499893856601} a^{19} + \frac{8139479674162566577964}{16490760440499893856601} a^{18} - \frac{18334434030141108373}{867934760026310202979} a^{17} - \frac{7769673497153588013301}{16490760440499893856601} a^{16} + \frac{1029437285495036375183}{16490760440499893856601} a^{15} + \frac{4333158049977057743356}{16490760440499893856601} a^{14} - \frac{8185153460066129507288}{16490760440499893856601} a^{13} - \frac{861531951845874832661}{16490760440499893856601} a^{12} + \frac{1704767596288743760456}{16490760440499893856601} a^{11} - \frac{973534432975094954086}{16490760440499893856601} a^{10} - \frac{7062731459198490927535}{16490760440499893856601} a^{9} - \frac{2003927410226776532026}{16490760440499893856601} a^{8} - \frac{750489420318848305671}{16490760440499893856601} a^{7} + \frac{6827601543896349025475}{16490760440499893856601} a^{6} - \frac{6671970860132147466961}{16490760440499893856601} a^{5} - \frac{7330972948669985280820}{16490760440499893856601} a^{4} - \frac{474917584159504700968}{16490760440499893856601} a^{3} + \frac{558218868749461884076}{16490760440499893856601} a^{2} + \frac{342261380158268158333}{16490760440499893856601} a + \frac{6453497947588683513811}{16490760440499893856601}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $19$ | 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: | \( 40310101291.2 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_2^4:D_5$ (as 20T87):
| A solvable group of order 320 |
| The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$ |
| Character table for $C_2\times C_2^4:D_5$ |
Intermediate fields
| 5.5.160801.1, 10.10.865923787055889.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | 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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $61$ | 61.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 61.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 61.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 61.8.4.1 | $x^{8} + 14884 x^{4} - 226981 x^{2} + 55383364$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 401 | Data not computed | ||||||