Normalized defining polynomial
\( x^{20} - 7440 x^{16} - 1454675 x^{14} + 266205060 x^{12} + 4959064482 x^{10} + 70829063500 x^{8} + 3416821527060 x^{6} + 84852613700100 x^{4} + 92497694344560 x^{2} + 231470329794201 \)
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: | \(426668469237698387780255274047851562500000000000000000000=2^{20}\cdot 5^{34}\cdot 31^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $678.43$ | 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: | $2, 5, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3100=2^{2}\cdot 5^{2}\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3100}(1,·)$, $\chi_{3100}(709,·)$, $\chi_{3100}(1799,·)$, $\chi_{3100}(841,·)$, $\chi_{3100}(2571,·)$, $\chi_{3100}(1549,·)$, $\chi_{3100}(2851,·)$, $\chi_{3100}(29,·)$, $\chi_{3100}(159,·)$, $\chi_{3100}(481,·)$, $\chi_{3100}(419,·)$, $\chi_{3100}(1511,·)$, $\chi_{3100}(39,·)$, $\chi_{3100}(1961,·)$, $\chi_{3100}(1131,·)$, $\chi_{3100}(1069,·)$, $\chi_{3100}(1391,·)$, $\chi_{3100}(1521,·)$, $\chi_{3100}(2689,·)$, $\chi_{3100}(2079,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{8} - \frac{1}{9} a^{4} - \frac{7}{27} a^{2} + \frac{1}{3}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{2511} a^{10} + \frac{1}{27} a^{6} + \frac{11}{81} a^{4} + \frac{2}{27} a^{2} + \frac{1}{3}$, $\frac{1}{2511} a^{11} + \frac{1}{27} a^{7} + \frac{11}{81} a^{5} + \frac{2}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{429381} a^{12} - \frac{49}{429381} a^{10} - \frac{46}{4617} a^{8} - \frac{199}{13851} a^{6} + \frac{1717}{13851} a^{4} - \frac{332}{1539} a^{2} + \frac{46}{171}$, $\frac{1}{429381} a^{13} - \frac{49}{429381} a^{11} + \frac{11}{4617} a^{9} - \frac{712}{13851} a^{7} + \frac{2230}{13851} a^{5} - \frac{1}{171} a^{3} + \frac{8}{171} a$, $\frac{1}{429381} a^{14} - \frac{10}{429381} a^{10} + \frac{221}{13851} a^{8} - \frac{113}{4617} a^{6} + \frac{604}{13851} a^{4} + \frac{619}{1539} a^{2} - \frac{26}{171}$, $\frac{1}{94034439} a^{15} + \frac{61}{94034439} a^{13} + \frac{18034}{94034439} a^{11} + \frac{3602}{3033369} a^{9} + \frac{102947}{3033369} a^{7} + \frac{353633}{3033369} a^{5} - \frac{12565}{337041} a^{3} - \frac{3262}{37449} a$, $\frac{1}{48239667207} a^{16} + \frac{38386}{48239667207} a^{14} + \frac{48694}{48239667207} a^{12} - \frac{5219236}{48239667207} a^{10} + \frac{12093635}{1556118297} a^{8} - \frac{7168141}{1556118297} a^{6} - \frac{2395571}{57634011} a^{4} - \frac{2393200}{6403779} a^{2} - \frac{4994}{29241}$, $\frac{1}{12011677134543} a^{17} + \frac{46081}{12011677134543} a^{15} + \frac{11865136}{12011677134543} a^{13} + \frac{461268593}{12011677134543} a^{11} - \frac{258133219}{387473455953} a^{9} - \frac{19224328717}{387473455953} a^{7} + \frac{859823707}{14350868739} a^{5} + \frac{193425056}{1594540971} a^{3} - \frac{195616466}{531513657} a$, $\frac{1}{272024290343258695588909816436223120635096343} a^{18} - \frac{440710344210509744030267826094868}{272024290343258695588909816436223120635096343} a^{16} + \frac{85945421339942681704999974783513900571}{272024290343258695588909816436223120635096343} a^{14} - \frac{250214772304899614128129550023057016278}{272024290343258695588909816436223120635096343} a^{12} + \frac{23193226566890965729839594344334192390791}{272024290343258695588909816436223120635096343} a^{10} - \frac{143219410499152549562993169622643128890301}{8774977107847054696416445691491068407583753} a^{8} + \frac{79013493197964512531966973031419279760726}{2924992369282351565472148563830356135861251} a^{6} - \frac{613077854233826246320785616870799113519}{5701739511271640478503213574718043149827} a^{4} - \frac{4261371052279360614898611761359998249805}{12037005634906796565729006435515868871857} a^{2} + \frac{169660661473739042464310757661537501}{662210795780755711378610685785105841}$, $\frac{1}{8432753000641019563256204309522916739687986633} a^{19} + \frac{3540600814030887828249589824662}{90674763447752898529636605478741040211698781} a^{17} - \frac{145414036659412880584095311314869293}{30224921149250966176545535159580346737232927} a^{15} - \frac{52568006254345537696682360381384123879}{272024290343258695588909816436223120635096343} a^{13} - \frac{2549091088479298127098563751092838874075}{90674763447752898529636605478741040211698781} a^{11} - \frac{14990191602151871481151568213494988375875}{10074973716416988725515178386526782245744309} a^{9} + \frac{4832240989309007979754925259389006930843}{105722615757193430077306574596277932621491} a^{7} - \frac{950680291151262098889886624640280263761}{17105218533814921435509640724154129449481} a^{5} - \frac{5666537047276830778795299162703683814379}{36111016904720389697187019306547606615571} a^{3} - \frac{208530494275389636499710520657085488351}{12037005634906796565729006435515868871857} a$
Class group and class number
$C_{1051348820}$, which has order $1051348820$ (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: | \( 410236188355566.6 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{10}$ (as 20T3):
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.1.0.1}{1} }^{20}$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | ||||||
| $5$ | 5.10.17.1 | $x^{10} - 5 x^{8} + 5$ | $10$ | $1$ | $17$ | $C_{10}$ | $[2]_{2}$ |
| 5.10.17.1 | $x^{10} - 5 x^{8} + 5$ | $10$ | $1$ | $17$ | $C_{10}$ | $[2]_{2}$ | |
| 31 | Data not computed | ||||||