Normalized defining polynomial
\( x^{20} - 4 x^{19} + 7 x^{18} - 6 x^{17} - 2 x^{16} + 6 x^{15} + 14 x^{14} - 54 x^{13} + 64 x^{12} + 24 x^{11} - 190 x^{10} + 290 x^{9} - 169 x^{8} - 154 x^{7} + 464 x^{6} - 554 x^{5} + 417 x^{4} - 208 x^{3} + 66 x^{2} - 12 x + 1 \)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(2500000000000000000000\)\(\medspace = 2^{20}\cdot 5^{22}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $11.75$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $2, 5$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $10$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{15} - \frac{2}{7} a^{14} - \frac{3}{7} a^{13} + \frac{2}{7} a^{11} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{17} - \frac{3}{7} a^{15} + \frac{2}{7} a^{14} - \frac{3}{7} a^{13} + \frac{2}{7} a^{12} + \frac{2}{7} a^{11} - \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{18} - \frac{1}{7} a^{15} - \frac{2}{7} a^{14} + \frac{2}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{22230187} a^{19} + \frac{316742}{22230187} a^{18} - \frac{1248133}{22230187} a^{17} + \frac{72912}{3175741} a^{16} - \frac{5356625}{22230187} a^{15} - \frac{2276879}{22230187} a^{14} + \frac{369525}{3175741} a^{13} + \frac{9538906}{22230187} a^{12} + \frac{10332540}{22230187} a^{11} + \frac{1228161}{3175741} a^{10} - \frac{222459}{22230187} a^{9} - \frac{6008298}{22230187} a^{8} + \frac{9247629}{22230187} a^{7} - \frac{1567752}{22230187} a^{6} - \frac{4433063}{22230187} a^{5} + \frac{8436821}{22230187} a^{4} + \frac{9943944}{22230187} a^{3} + \frac{1386475}{22230187} a^{2} - \frac{109510}{22230187} a - \frac{201610}{3175741}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -\frac{57119}{9107} a^{19} + \frac{26866}{1301} a^{18} - \frac{267410}{9107} a^{17} + \frac{156983}{9107} a^{16} + \frac{218307}{9107} a^{15} - \frac{181352}{9107} a^{14} - \frac{926865}{9107} a^{13} + \frac{2419212}{9107} a^{12} - \frac{1951750}{9107} a^{11} - \frac{2701613}{9107} a^{10} + \frac{1266459}{1301} a^{9} - \frac{10294686}{9107} a^{8} + \frac{2557279}{9107} a^{7} + \frac{10292841}{9107} a^{6} - \frac{19014662}{9107} a^{5} + \frac{18350921}{9107} a^{4} - \frac{11338829}{9107} a^{3} + \frac{4435672}{9107} a^{2} - \frac{1011806}{9107} a + \frac{107027}{9107} \) (order $4$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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 | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 397.737246993 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_{10}\times D_5$ (as 20T24):
| A solvable group of order 100 |
| The 40 conjugacy class representatives for $C_{10}\times D_5$ |
| Character table for $C_{10}\times D_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), 10.0.400000000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $5$ | 5.10.17.5 | $x^{10} - 5 x^{8} + 55$ | $10$ | $1$ | $17$ | $C_{10}$ | $[2]_{2}$ |
| 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |