Normalized defining polynomial
\( x^{20} + 8 x^{18} + 24 x^{16} + x^{14} - 170 x^{12} - 403 x^{10} - 340 x^{8} + 4 x^{6} + 192 x^{4} + 128 x^{2} + 32 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(389963020402083454798266368=2^{25}\cdot 7^{8}\cdot 17^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $21.36$ | 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: | $2, 7, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{3}{7} a^{8} - \frac{3}{7} a^{6} + \frac{2}{7} a^{4} - \frac{3}{7} a^{2} + \frac{2}{7}$, $\frac{1}{7} a^{11} - \frac{3}{7} a^{9} - \frac{3}{7} a^{7} + \frac{2}{7} a^{5} - \frac{3}{7} a^{3} + \frac{2}{7} a$, $\frac{1}{14} a^{12} + \frac{1}{7} a^{8} - \frac{1}{2} a^{6} - \frac{2}{7} a^{4} - \frac{1}{2} a^{2} + \frac{3}{7}$, $\frac{1}{14} a^{13} + \frac{1}{7} a^{9} - \frac{1}{2} a^{7} - \frac{2}{7} a^{5} - \frac{1}{2} a^{3} + \frac{3}{7} a$, $\frac{1}{28} a^{14} + \frac{13}{28} a^{8} + \frac{1}{14} a^{6} - \frac{11}{28} a^{4} + \frac{3}{7} a^{2} - \frac{1}{7}$, $\frac{1}{28} a^{15} + \frac{13}{28} a^{9} + \frac{1}{14} a^{7} - \frac{11}{28} a^{5} + \frac{3}{7} a^{3} - \frac{1}{7} a$, $\frac{1}{392} a^{16} - \frac{3}{196} a^{14} + \frac{1}{49} a^{12} - \frac{15}{392} a^{10} - \frac{4}{49} a^{8} + \frac{89}{392} a^{6} + \frac{51}{196} a^{4} + \frac{23}{98} a^{2} + \frac{23}{49}$, $\frac{1}{392} a^{17} - \frac{3}{196} a^{15} + \frac{1}{49} a^{13} - \frac{15}{392} a^{11} - \frac{4}{49} a^{9} + \frac{89}{392} a^{7} + \frac{51}{196} a^{5} + \frac{23}{98} a^{3} + \frac{23}{49} a$, $\frac{1}{784} a^{18} - \frac{23}{784} a^{12} + \frac{23}{392} a^{10} + \frac{37}{784} a^{8} + \frac{47}{196} a^{6} + \frac{43}{196} a^{4} - \frac{27}{98} a^{2} + \frac{13}{49}$, $\frac{1}{784} a^{19} - \frac{23}{784} a^{13} + \frac{23}{392} a^{11} + \frac{37}{784} a^{9} + \frac{47}{196} a^{7} + \frac{43}{196} a^{5} - \frac{27}{98} a^{3} + \frac{13}{49} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 219934.649034 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$D_4\times D_5$ (as 20T21):
| A solvable group of order 80 |
| The 20 conjugacy class representatives for $D_4\times D_5$ |
| Character table for $D_4\times D_5$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.9248.1, 5.1.14161.1, 10.2.3409076657.1 |
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 | R | $20$ | $20$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.15.9 | $x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
| 2.10.10.7 | $x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $17$ | 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |