Normalized defining polynomial
\( x^{20} - 2 x^{19} + 10 x^{18} - 10 x^{17} - 19 x^{16} - 44 x^{15} + 40 x^{14} - 256 x^{13} - 1019 x^{12} - 1170 x^{11} + 2672 x^{10} + 4996 x^{9} - 604 x^{8} + 1902 x^{7} + 24450 x^{6} + 43868 x^{5} + 48635 x^{4} + 28394 x^{3} + 7652 x^{2} + 1192 x + 89 \)
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: | \(18655986742119776375747379200000=2^{30}\cdot 5^{5}\cdot 11^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.61$ | 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}{14920592616944053965323999212808146802760753613} a^{19} + \frac{152456014573118779228077395996361685323750538}{346990525975443115472651144483910390761877991} a^{18} + \frac{5546528249536965240374294421569046187608502894}{14920592616944053965323999212808146802760753613} a^{17} - \frac{15909848562386505829644104002428873853686477}{14920592616944053965323999212808146802760753613} a^{16} - \frac{6721882903145818220940403389801264335709309053}{14920592616944053965323999212808146802760753613} a^{15} + \frac{817785228629272106950874009425294533795725029}{14920592616944053965323999212808146802760753613} a^{14} + \frac{6971069659316127309907928361612318640636337472}{14920592616944053965323999212808146802760753613} a^{13} - \frac{147704695846233761699041170351134443410665431}{346990525975443115472651144483910390761877991} a^{12} - \frac{4288748777839352154037666010781716426927712539}{14920592616944053965323999212808146802760753613} a^{11} + \frac{2940683755083719095890732720133699652830320709}{14920592616944053965323999212808146802760753613} a^{10} + \frac{3622042400884350238395581937476686192796626422}{14920592616944053965323999212808146802760753613} a^{9} + \frac{1845330812508339819667491371696924552436826303}{14920592616944053965323999212808146802760753613} a^{8} - \frac{7414833328290378618888587420604995705500704219}{14920592616944053965323999212808146802760753613} a^{7} - \frac{4795222134018703383159087599722057071663477324}{14920592616944053965323999212808146802760753613} a^{6} + \frac{4446540390411416565530131866857013886238849479}{14920592616944053965323999212808146802760753613} a^{5} - \frac{2610326052478591231067464147344483298430078446}{14920592616944053965323999212808146802760753613} a^{4} - \frac{5690900757154875332454528735345527431000531506}{14920592616944053965323999212808146802760753613} a^{3} + \frac{1686016713300660053942030583049516882659456047}{14920592616944053965323999212808146802760753613} a^{2} + \frac{3174608313296016628829089851115670701330030025}{14920592616944053965323999212808146802760753613} a + \frac{39557933388884826098510029754495306763727610}{167647108055551168149707856323687042727648917}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 40370967.7281 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 640 |
| The 40 conjugacy class representatives for t20n130 |
| Character table for t20n130 is not computed |
Intermediate fields
| \(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\) |
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 | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ | R | $20$ | $20$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ | $20$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | ||||||
| $5$ | 5.5.0.1 | $x^{5} - x + 2$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 5.5.0.1 | $x^{5} - x + 2$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 11 | Data not computed | ||||||