Normalized defining polynomial
\( x^{20} - 3 x^{19} + 2 x^{18} + 3 x^{17} - 2 x^{16} - 9 x^{15} + 18 x^{14} - 13 x^{13} - x^{12} + 6 x^{11} + 10 x^{10} - 26 x^{9} + 20 x^{8} - 6 x^{7} + x^{6} + 5 x^{5} - 10 x^{4} + 6 x^{3} - 2 x + 1 \)
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: | \(3575480621700351753081\)\(\medspace = 3^{10}\cdot 7^{10}\cdot 11^{8}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
Root discriminant: | $11.96$ | 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: | $3, 7, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2663593} a^{19} - \frac{605577}{2663593} a^{18} + \frac{865553}{2663593} a^{17} - \frac{1243914}{2663593} a^{16} - \frac{768917}{2663593} a^{15} + \frac{133054}{2663593} a^{14} - \frac{354728}{2663593} a^{13} + \frac{605595}{2663593} a^{12} - \frac{1111512}{2663593} a^{11} - \frac{501171}{2663593} a^{10} + \frac{1013558}{2663593} a^{9} + \frac{680637}{2663593} a^{8} - \frac{1035426}{2663593} a^{7} + \frac{1290760}{2663593} a^{6} - \frac{685238}{2663593} a^{5} + \frac{1163147}{2663593} a^{4} - \frac{394096}{2663593} a^{3} - \frac{978097}{2663593} a^{2} - \frac{1053511}{2663593} a + \frac{402138}{2663593}$
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: | \( -\frac{5069577}{2663593} a^{19} + \frac{15210989}{2663593} a^{18} - \frac{13772404}{2663593} a^{17} - \frac{5212947}{2663593} a^{16} + \frac{3481178}{2663593} a^{15} + \frac{38083464}{2663593} a^{14} - \frac{89138870}{2663593} a^{13} + \frac{95969693}{2663593} a^{12} - \frac{52519168}{2663593} a^{11} + \frac{16837722}{2663593} a^{10} - \frac{63630828}{2663593} a^{9} + \frac{132396579}{2663593} a^{8} - \frac{146687436}{2663593} a^{7} + \frac{116295184}{2663593} a^{6} - \frac{79084843}{2663593} a^{5} + \frac{20213325}{2663593} a^{4} + \frac{18815882}{2663593} a^{3} - \frac{27982575}{2663593} a^{2} + \frac{18220722}{2663593} a - \frac{3418100}{2663593} \) (order $6$) | 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: | \( 635.552521857 \) | 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
\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), 10.0.246071287.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | R | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
3 | Data not computed | ||||||
$7$ | 7.10.5.2 | $x^{10} - 2401 x^{2} + 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
7.10.5.2 | $x^{10} - 2401 x^{2} + 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
$11$ | 11.10.0.1 | $x^{10} + x^{2} - x + 6$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |
11.10.8.1 | $x^{10} + 220 x^{5} + 41503$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |