Normalized defining polynomial
\( x^{20} - 6 x^{19} + 17 x^{18} - 26 x^{17} + 3 x^{16} + 99 x^{15} - 278 x^{14} + 401 x^{13} - 222 x^{12} - 386 x^{11} + 1130 x^{10} - 1293 x^{9} + 290 x^{8} + 1647 x^{7} - 3426 x^{6} + 3993 x^{5} - 3230 x^{4} + 1866 x^{3} - 736 x^{2} + 176 x - 19 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-583135620108095986484224=-\,2^{10}\cdot 7^{10}\cdot 17^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.43$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} + \frac{5}{13} a^{17} - \frac{1}{13} a^{16} + \frac{1}{13} a^{15} - \frac{4}{13} a^{14} - \frac{5}{13} a^{13} - \frac{2}{13} a^{12} + \frac{3}{13} a^{11} - \frac{4}{13} a^{10} + \frac{1}{13} a^{9} + \frac{3}{13} a^{8} + \frac{6}{13} a^{7} - \frac{6}{13} a^{6} - \frac{1}{13} a^{5} + \frac{4}{13} a^{4} + \frac{2}{13} a^{3} - \frac{3}{13} a^{2} - \frac{3}{13} a - \frac{4}{13}$, $\frac{1}{11688761593144519} a^{19} - \frac{214737302820402}{11688761593144519} a^{18} - \frac{1963937329352729}{11688761593144519} a^{17} + \frac{3436609733702344}{11688761593144519} a^{16} - \frac{1520554966730248}{11688761593144519} a^{15} - \frac{3834582586453352}{11688761593144519} a^{14} - \frac{226716329560583}{899135507164963} a^{13} + \frac{5638916628449399}{11688761593144519} a^{12} - \frac{367320727899111}{899135507164963} a^{11} + \frac{446071022335612}{899135507164963} a^{10} - \frac{1350410125452}{899135507164963} a^{9} - \frac{2957745480184758}{11688761593144519} a^{8} - \frac{5320516036993103}{11688761593144519} a^{7} - \frac{941340621842071}{11688761593144519} a^{6} + \frac{3025577732112878}{11688761593144519} a^{5} - \frac{3440951243620628}{11688761593144519} a^{4} + \frac{1644493904715341}{11688761593144519} a^{3} - \frac{4288443356201245}{11688761593144519} a^{2} + \frac{2279355359862950}{11688761593144519} a - \frac{5278695494601218}{11688761593144519}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $10$ | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 5497.47001539 \) | 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
| \(\Q(\sqrt{17}) \), 4.2.56644.1, 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 | $20$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ | $20$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.0.1 | $x^{10} - x^{3} + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |
| 2.10.10.11 | $x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| $7$ | 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $17$ | 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 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |