Normalized defining polynomial
\( x^{20} - 2 x^{19} + 2 x^{18} + 2 x^{17} - 5 x^{16} + 4 x^{15} + 4 x^{14} - 12 x^{13} + 7 x^{12} + 2 x^{11} - 14 x^{10} + 6 x^{9} + 5 x^{8} - 12 x^{7} + 8 x^{6} + 4 x^{5} - 4 x^{4} + 8 x^{3} + 4 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2502343454824177132896256=2^{32}\cdot 17^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.59$ | 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: | $2, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} + \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{8}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{9}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{10}$, $\frac{1}{164} a^{19} + \frac{1}{41} a^{18} - \frac{15}{164} a^{17} - \frac{3}{82} a^{16} + \frac{1}{41} a^{14} - \frac{13}{164} a^{13} - \frac{2}{41} a^{12} - \frac{1}{4} a^{11} + \frac{1}{82} a^{10} + \frac{39}{164} a^{9} - \frac{3}{82} a^{8} + \frac{5}{82} a^{7} - \frac{17}{82} a^{6} - \frac{73}{164} a^{5} - \frac{6}{41} a^{4} - \frac{33}{82} a^{3} - \frac{15}{41} a^{2} + \frac{25}{82} a - \frac{7}{41}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{11}{41} a^{19} - \frac{111}{164} a^{18} + \frac{119}{164} a^{17} + \frac{23}{164} a^{16} - \frac{5}{4} a^{15} + \frac{44}{41} a^{14} + \frac{21}{41} a^{13} - \frac{129}{41} a^{12} + \frac{5}{2} a^{11} - \frac{35}{164} a^{10} - \frac{375}{164} a^{9} + \frac{351}{164} a^{8} + \frac{317}{164} a^{7} - \frac{215}{82} a^{6} + \frac{99}{41} a^{5} + \frac{5}{82} a^{4} - \frac{99}{82} a^{3} + \frac{37}{41} a^{2} - \frac{24}{41} a - \frac{21}{41} \) (order $4$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 30574.1646172 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times A_5$ (as 20T31):
| A non-solvable group of order 120 |
| The 10 conjugacy class representatives for $C_2\times A_5$ |
| Character table for $C_2\times A_5$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 10.2.98867482624.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 sibling: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 20 sibling: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 40 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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.14.2 | $x^{8} + 2 x^{7} + 2$ | $8$ | $1$ | $14$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ |
| 2.12.18.51 | $x^{12} + 10 x^{11} + 16 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} - 8 x^{6} - 8 x^{5} + 4 x^{4} - 8 x^{3} + 16 x + 8$ | $4$ | $3$ | $18$ | $A_4 \times C_2$ | $[2, 2, 2]^{3}$ | |
| $17$ | $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 17.3.2.1 | $x^{3} - 17$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 17.3.2.1 | $x^{3} - 17$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 17.6.4.1 | $x^{6} + 136 x^{3} + 7803$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 17.6.4.1 | $x^{6} + 136 x^{3} + 7803$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |