Normalized defining polynomial
\( x^{20} - 3 x^{19} - 30 x^{18} + 57 x^{17} + 417 x^{16} - 393 x^{15} - 4122 x^{14} + 4795 x^{13} + 35632 x^{12} - 47393 x^{11} - 236955 x^{10} + 66599 x^{9} + 731100 x^{8} + 523933 x^{7} - 352102 x^{6} - 432661 x^{5} + 136989 x^{4} + 81080 x^{3} - 47110 x^{2} + 10050 x - 775 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(474764677846779705681219512939453125=5^{15}\cdot 11^{5}\cdot 9931^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $60.79$ | 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: | $5, 11, 9931$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{545} a^{18} + \frac{117}{545} a^{17} - \frac{18}{109} a^{16} + \frac{172}{545} a^{15} + \frac{42}{545} a^{14} - \frac{208}{545} a^{13} + \frac{18}{545} a^{12} + \frac{5}{109} a^{11} + \frac{107}{545} a^{10} - \frac{103}{545} a^{9} + \frac{31}{109} a^{8} + \frac{54}{545} a^{7} + \frac{29}{109} a^{6} + \frac{113}{545} a^{5} + \frac{243}{545} a^{4} + \frac{79}{545} a^{3} - \frac{21}{545} a^{2} + \frac{32}{109} a + \frac{10}{109}$, $\frac{1}{1323362094548037129790090713268857364044241269439211075} a^{19} + \frac{641978283961656838514355035914294969390383165438812}{1323362094548037129790090713268857364044241269439211075} a^{18} + \frac{70081833612683480135526407009681714645967345514529684}{264672418909607425958018142653771472808848253887842215} a^{17} + \frac{341637295968561542240561589303256920818005669850825147}{1323362094548037129790090713268857364044241269439211075} a^{16} - \frac{574963102557199431600570244114254855379015031226682653}{1323362094548037129790090713268857364044241269439211075} a^{15} - \frac{135786985018436374093732736115853706546464902799712673}{1323362094548037129790090713268857364044241269439211075} a^{14} - \frac{88384707275118234725077957257510703100040640047329452}{1323362094548037129790090713268857364044241269439211075} a^{13} + \frac{31726643145857845952339150794037717096751772012964286}{264672418909607425958018142653771472808848253887842215} a^{12} - \frac{639012946924193630302633089779439286398508255968926383}{1323362094548037129790090713268857364044241269439211075} a^{11} - \frac{519826093881351698395685032884985600769739803467784563}{1323362094548037129790090713268857364044241269439211075} a^{10} - \frac{24289030620739044255522571093356480962681142509914552}{264672418909607425958018142653771472808848253887842215} a^{9} + \frac{645844267378643887623299053294808638974663515654150889}{1323362094548037129790090713268857364044241269439211075} a^{8} - \frac{1975172452968046520918505089703250025428322855183058}{264672418909607425958018142653771472808848253887842215} a^{7} + \frac{88888593173895869910237872217610646511275997013167513}{1323362094548037129790090713268857364044241269439211075} a^{6} + \frac{130421641250532866970633050964898476849828231107491693}{1323362094548037129790090713268857364044241269439211075} a^{5} - \frac{317275296405317908248731988330552916684802842913436506}{1323362094548037129790090713268857364044241269439211075} a^{4} - \frac{437058085966161029775340358964062365509187065241749816}{1323362094548037129790090713268857364044241269439211075} a^{3} - \frac{13277370091677456977233997925137844455759045721094631}{264672418909607425958018142653771472808848253887842215} a^{2} - \frac{89929630125037803168951326429177721223655481416861121}{264672418909607425958018142653771472808848253887842215} a - \frac{14332235366788819262205980401624734178743614296416489}{52934483781921485191603628530754294561769650777568443}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 81096966325.5 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 7372800 |
| The 324 conjugacy class representatives for t20n1023 are not computed |
| Character table for t20n1023 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.10.932312193828125.1 |
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 | $20$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.12.11.2 | $x^{12} - 20$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $[\ ]_{12}^{2}$ | |
| $11$ | 11.2.1.2 | $x^{2} + 33$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 11.10.0.1 | $x^{10} + x^{2} - x + 6$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 9931 | Data not computed | ||||||