Normalized defining polynomial
\( x^{20} - 3 x^{19} - 3 x^{18} + 18 x^{17} - 32 x^{16} + 19 x^{15} + 6 x^{14} - 95 x^{13} + 112 x^{12} + 21 x^{11} - 169 x^{10} - 21 x^{9} + 112 x^{8} + 95 x^{7} + 6 x^{6} - 19 x^{5} - 32 x^{4} - 18 x^{3} - 3 x^{2} + 3 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(237377022439742902041015625=5^{10}\cdot 11^{16}\cdot 23^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $20.83$ | 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: | $5, 11, 23$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{71040859} a^{18} + \frac{28128191}{71040859} a^{17} + \frac{24075137}{71040859} a^{16} - \frac{16254249}{71040859} a^{15} - \frac{15994771}{71040859} a^{14} - \frac{4573316}{71040859} a^{13} + \frac{17871374}{71040859} a^{12} + \frac{18568746}{71040859} a^{11} - \frac{17108128}{71040859} a^{10} - \frac{33728043}{71040859} a^{9} + \frac{17108128}{71040859} a^{8} + \frac{18568746}{71040859} a^{7} - \frac{17871374}{71040859} a^{6} - \frac{4573316}{71040859} a^{5} + \frac{15994771}{71040859} a^{4} - \frac{16254249}{71040859} a^{3} - \frac{24075137}{71040859} a^{2} + \frac{28128191}{71040859} a - \frac{1}{71040859}$, $\frac{1}{28203221023} a^{19} - \frac{66}{28203221023} a^{18} - \frac{4542669536}{28203221023} a^{17} + \frac{13856220122}{28203221023} a^{16} + \frac{1474337946}{28203221023} a^{15} - \frac{12009060854}{28203221023} a^{14} - \frac{7116125193}{28203221023} a^{13} - \frac{5535003231}{28203221023} a^{12} + \frac{7586066145}{28203221023} a^{11} - \frac{7783726544}{28203221023} a^{10} - \frac{946998525}{28203221023} a^{9} + \frac{3861320002}{28203221023} a^{8} - \frac{134194851}{655888861} a^{7} - \frac{12424338666}{28203221023} a^{6} - \frac{4276367436}{28203221023} a^{5} + \frac{301925705}{655888861} a^{4} + \frac{2176666170}{28203221023} a^{3} + \frac{150703043}{28203221023} a^{2} - \frac{9610645663}{28203221023} a - \frac{5299936168}{28203221023}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 284529.931338 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2\times C_2^4:C_5$ (as 20T86):
| A solvable group of order 320 |
| The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$ |
| Character table for $C_2^2\times C_2^4:C_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.4.4930254263.1, 10.10.669871503125.1, 10.4.15407044571875.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 | Data not computed | ||||||
| $11$ | 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
| $23$ | 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |