Normalized defining polynomial
\( x^{20} - 3 x^{19} + x^{18} - x^{17} + 14 x^{16} - 8 x^{15} - 9 x^{14} - 10 x^{13} + 6 x^{12} + 15 x^{11} - 24 x^{10} + 16 x^{9} + 31 x^{8} - 29 x^{7} + 3 x^{6} + 7 x^{5} - 24 x^{4} - 7 x^{3} + 3 x^{2} - 2 x + 1 \)
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: | \(-132745109443598544921875=-\,5^{10}\cdot 7^{8}\cdot 11^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.33$ | 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: | $5, 7, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{665} a^{18} - \frac{55}{133} a^{17} - \frac{36}{133} a^{16} - \frac{131}{665} a^{15} + \frac{141}{665} a^{14} + \frac{6}{665} a^{13} + \frac{29}{95} a^{12} + \frac{288}{665} a^{11} + \frac{26}{95} a^{10} - \frac{262}{665} a^{9} + \frac{8}{665} a^{8} + \frac{92}{665} a^{7} + \frac{37}{95} a^{6} - \frac{194}{665} a^{5} + \frac{8}{35} a^{4} - \frac{3}{665} a^{3} - \frac{43}{133} a^{2} + \frac{18}{95} a + \frac{296}{665}$, $\frac{1}{6190894610075} a^{19} - \frac{3585632264}{6190894610075} a^{18} + \frac{314453767793}{1238178922015} a^{17} + \frac{2802136393334}{6190894610075} a^{16} + \frac{431577319038}{1238178922015} a^{15} - \frac{1232604267058}{6190894610075} a^{14} - \frac{1275812035111}{6190894610075} a^{13} - \frac{277173694804}{6190894610075} a^{12} + \frac{601116123301}{1238178922015} a^{11} - \frac{262428988642}{1238178922015} a^{10} - \frac{19277750043}{46548079775} a^{9} + \frac{12260430859}{65167311685} a^{8} + \frac{1849718940606}{6190894610075} a^{7} + \frac{96828277459}{247635784403} a^{6} + \frac{1576845572093}{6190894610075} a^{5} - \frac{202940221931}{6190894610075} a^{4} + \frac{25485132823}{325836558425} a^{3} + \frac{1319584358256}{6190894610075} a^{2} + \frac{2594019367837}{6190894610075} a + \frac{153084089476}{6190894610075}$
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: | \( 2389.27705771 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_{10}^2:C_2^2$ (as 20T106):
| A solvable group of order 400 |
| The 46 conjugacy class representatives for $C_{10}^2:C_2^2$ |
| Character table for $C_{10}^2:C_2^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.275.1, 10.2.109853253125.1 |
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 | $20$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | R | R | $20$ | $20$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | $20$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $11$ | 11.10.9.1 | $x^{10} - 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.0.1 | $x^{10} + x^{2} - x + 6$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |