Normalized defining polynomial
\( x^{20} + 87 x^{18} + 3799 x^{16} + 61393 x^{14} - 87464 x^{12} - 13292005 x^{10} - 63045565 x^{8} + 1159940840 x^{6} + 7631561990 x^{4} - 76711697260 x^{2} + 25126157525 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(34516754990392737763796000000000000000000=2^{20}\cdot 5^{18}\cdot 29^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $106.39$ | 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: | $2, 5, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{29} a^{4}$, $\frac{1}{29} a^{5}$, $\frac{1}{29} a^{6}$, $\frac{1}{29} a^{7}$, $\frac{1}{841} a^{8}$, $\frac{1}{841} a^{9}$, $\frac{1}{841} a^{10}$, $\frac{1}{841} a^{11}$, $\frac{1}{24389} a^{12}$, $\frac{1}{24389} a^{13}$, $\frac{1}{24389} a^{14}$, $\frac{1}{24389} a^{15}$, $\frac{1}{338787599} a^{16} - \frac{107}{11682331} a^{14} + \frac{168}{11682331} a^{12} + \frac{144}{402839} a^{10} - \frac{69}{402839} a^{8} + \frac{26}{13891} a^{6} - \frac{123}{13891} a^{4} + \frac{178}{479} a^{2} + \frac{5}{479}$, $\frac{1}{338787599} a^{17} - \frac{107}{11682331} a^{15} + \frac{168}{11682331} a^{13} + \frac{144}{402839} a^{11} - \frac{69}{402839} a^{9} + \frac{26}{13891} a^{7} - \frac{123}{13891} a^{5} + \frac{178}{479} a^{3} + \frac{5}{479} a$, $\frac{1}{770274825394368338903478005} a^{18} + \frac{5338391558386337}{770274825394368338903478005} a^{16} - \frac{5687759634684316613}{1154834820681211902404015} a^{14} - \frac{373379919052340185823}{26561200875667873755292345} a^{12} + \frac{419674561281170351576}{915903478471305991561805} a^{10} - \frac{45173405718238734186}{183180695694261198312361} a^{8} - \frac{70211464589793684550}{6316575713595213734909} a^{6} + \frac{22982447976326043485}{6316575713595213734909} a^{4} - \frac{71640912613053073619}{217812955641214266721} a^{2} - \frac{35467836173497772999}{217812955641214266721}$, $\frac{1}{5391923777760578372324346035} a^{19} + \frac{5338391558386337}{5391923777760578372324346035} a^{17} - \frac{53038402165383070248}{8083843744768483316828105} a^{15} + \frac{715684859153731147782}{185928406129675116287046415} a^{13} + \frac{52025494465077299489}{221080149975832480721815} a^{11} - \frac{480799317000667267628}{1282264869859828388186527} a^{9} + \frac{583227402333849115613}{44216029995166496144363} a^{7} + \frac{458608359258754576927}{44216029995166496144363} a^{5} + \frac{581797954310589726544}{1524690689488499867047} a^{3} - \frac{253280791814712039720}{1524690689488499867047} a$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 1565634747930 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_5:D_5.Q_8$ (as 20T105):
| A solvable group of order 400 |
| The 28 conjugacy class representatives for $C_5:D_5.Q_8$ |
| Character table for $C_5:D_5.Q_8$ is not computed |
Intermediate fields
| \(\Q(\sqrt{29}) \), 4.4.9755600.2, 10.2.8012167578125.1 |
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 | R | $20$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ | $20$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | $20$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | R | $20$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | $20$ | $20$ | $20$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$ |
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 | ||||||
| $5$ | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.10.11.8 | $x^{10} + 20 x^{2} + 10$ | $10$ | $1$ | $11$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ | |
| 29 | Data not computed | ||||||