Normalized defining polynomial
\( x^{20} - 2 x^{19} - 12 x^{18} - 6 x^{17} + 10 x^{16} + 26 x^{15} - 58 x^{14} - 98 x^{13} + 211 x^{12} + 174 x^{11} + 44 x^{10} - 446 x^{9} - 173 x^{8} + 682 x^{7} - 674 x^{6} + 284 x^{5} + 125 x^{4} - 100 x^{3} + 32 x^{2} - 18 x - 1 \)
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: | \(154181708612560135336755200000=2^{30}\cdot 5^{5}\cdot 11^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.80$ | 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: | $2, 5, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{25132852223710054579399117} a^{19} + \frac{14348184237584839709501}{584484935435117548358119} a^{18} - \frac{3484586274908769594079276}{25132852223710054579399117} a^{17} - \frac{9207718501016697940828991}{25132852223710054579399117} a^{16} - \frac{183042564968100185308072}{25132852223710054579399117} a^{15} + \frac{574492208347705536861325}{25132852223710054579399117} a^{14} - \frac{4838247622522930601120658}{25132852223710054579399117} a^{13} + \frac{12234562375801912615246625}{25132852223710054579399117} a^{12} + \frac{161409690432671247925202}{375117197368806784767151} a^{11} + \frac{11213667048074737095097978}{25132852223710054579399117} a^{10} - \frac{10209440967773364048125961}{25132852223710054579399117} a^{9} + \frac{4855740121410539041867570}{25132852223710054579399117} a^{8} - \frac{3138314052112171713574811}{25132852223710054579399117} a^{7} - \frac{5892227117950189006290124}{25132852223710054579399117} a^{6} + \frac{1760281344258933829348915}{25132852223710054579399117} a^{5} + \frac{9403075915767717038056580}{25132852223710054579399117} a^{4} - \frac{3269104541018208027124821}{25132852223710054579399117} a^{3} - \frac{80400592944360368654287}{25132852223710054579399117} a^{2} + \frac{7059642585196278782390939}{25132852223710054579399117} a + \frac{11893455012129827406896126}{25132852223710054579399117}$
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: | \( 10596758.6186 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 10240 |
| The 136 conjugacy class representatives for t20n427 are not computed |
| Character table for t20n427 is not computed |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.4.219503494144.2 |
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 | $20$ | R | $20$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ | $20$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | ||||||
| $5$ | 5.10.5.2 | $x^{10} - 625 x^{2} + 6250$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 5.10.0.1 | $x^{10} + x^{2} - x + 3$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| $11$ | 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |