Normalized defining polynomial
\( x^{20} - 2 x^{19} - 2 x^{18} - 16 x^{17} + 89 x^{16} - 184 x^{15} + 590 x^{14} - 1944 x^{13} + 4371 x^{12} - 8374 x^{11} + 16186 x^{10} - 29220 x^{9} + 46110 x^{8} - 65880 x^{7} + 84452 x^{6} - 100670 x^{5} + 116483 x^{4} - 111656 x^{3} + 75518 x^{2} - 30558 x + 8047 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(9964694368301019319239964622848=2^{30}\cdot 13^{7}\cdot 101^{2}\cdot 347^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $35.48$ | 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: | $2, 13, 101, 347$ | 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}{258928097072199509343343137679107099216830791} a^{19} + \frac{74644467283070445909745063867191384071732043}{258928097072199509343343137679107099216830791} a^{18} - \frac{21478954362349426744131328981796846921617351}{258928097072199509343343137679107099216830791} a^{17} + \frac{94751839676638205429436543617968763817807106}{258928097072199509343343137679107099216830791} a^{16} - \frac{126243152704632605669194401282443620609931958}{258928097072199509343343137679107099216830791} a^{15} - \frac{41525670849538646040660504545218130729922704}{258928097072199509343343137679107099216830791} a^{14} + \frac{62370308582105253498723110106865030006456017}{258928097072199509343343137679107099216830791} a^{13} - \frac{77967322069851921661384061325412852664890223}{258928097072199509343343137679107099216830791} a^{12} + \frac{14050813584539466128480086530081516496739253}{258928097072199509343343137679107099216830791} a^{11} + \frac{78259063338410439502243441020670957310892936}{258928097072199509343343137679107099216830791} a^{10} + \frac{83414025483375768818960547764855596553651766}{258928097072199509343343137679107099216830791} a^{9} + \frac{74248069026624751612928684225631248585675257}{258928097072199509343343137679107099216830791} a^{8} - \frac{62388945210824478049149543170399950003770239}{258928097072199509343343137679107099216830791} a^{7} + \frac{119195811514181584829544976541212794978715864}{258928097072199509343343137679107099216830791} a^{6} - \frac{59958044167688967847607114585039874575392519}{258928097072199509343343137679107099216830791} a^{5} + \frac{57443384377902082562597866043428178779433374}{258928097072199509343343137679107099216830791} a^{4} + \frac{6375619651594875872847761836464476392495386}{15231064533658794667255478687006299953931223} a^{3} + \frac{1049258249444190971921147278377542996083102}{15231064533658794667255478687006299953931223} a^{2} + \frac{1026796846889683264269567397391767160604863}{258928097072199509343343137679107099216830791} a - \frac{9933822373371218297328870003771301269614894}{258928097072199509343343137679107099216830791}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 6172515.52201 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 245760 |
| The 201 conjugacy class representatives for t20n886 are not computed |
| Character table for t20n886 is not computed |
Intermediate fields
| 5.3.4511.1, 10.6.2104587490304.1 |
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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ | $20$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 | ||||||
| $13$ | $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $101$ | $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 101.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 101.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 101.4.0.1 | $x^{4} - x + 12$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 101.4.0.1 | $x^{4} - x + 12$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 101.4.2.1 | $x^{4} + 505 x^{2} + 91809$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 347 | Data not computed | ||||||