Normalized defining polynomial
\( x^{21} - 4 x^{20} - 118 x^{19} + 291 x^{18} + 6254 x^{17} - 5793 x^{16} - 184349 x^{15} - 86077 x^{14} + 3074544 x^{13} + 5414884 x^{12} - 25253619 x^{11} - 86077434 x^{10} + 27446591 x^{9} + 505665846 x^{8} + 860543159 x^{7} + 166010847 x^{6} - 2164621649 x^{5} - 6780985888 x^{4} - 13378205010 x^{3} - 18177795513 x^{2} - 18348067445 x - 11878444475 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[9, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1123867680627116189703049190875255649932194273960649=7^{4}\cdot 577^{10}\cdot 10697503699^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $269.77$ | 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: | $7, 577, 10697503699$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{35} a^{19} + \frac{1}{5} a^{17} + \frac{4}{35} a^{16} + \frac{1}{7} a^{15} + \frac{12}{35} a^{14} + \frac{4}{35} a^{13} - \frac{1}{35} a^{12} + \frac{2}{7} a^{11} - \frac{16}{35} a^{10} - \frac{3}{35} a^{9} + \frac{9}{35} a^{8} - \frac{13}{35} a^{7} + \frac{9}{35} a^{6} + \frac{1}{5} a^{4} - \frac{1}{35} a^{3} - \frac{2}{35} a^{2} + \frac{12}{35} a + \frac{1}{7}$, $\frac{1}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a^{20} - \frac{207732334376973432974633959200081934921953142550248342201178445476452748251796792360286}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a^{19} + \frac{1309336078228612446530267312415094644453554584316629527302724002527301327827233939220901}{3473230923504514068298735536222404533389561491910001729073776071337665411477940044334085} a^{18} - \frac{8708385806877183354067646222062210297068935156733169979050024802648646638998815773058218}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a^{17} - \frac{9522256582697088787613068510028998823081509742652354874899854051496355431288560100449299}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a^{16} - \frac{9352716725398166894983503509030526014538384118192505137770281310362808672406696696113818}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a^{15} + \frac{11413753112719145915472522671737831686901615231917636792576708082511246705441101582376067}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a^{14} - \frac{1114874353709217906487554201729413118632158110270644835934767346180996863658257870643555}{4862523292906319695618229750711366346745386088674002420703286499872731576069116062067719} a^{13} + \frac{87912027825649494246958406616490497150259513049398647785364905893739446503055213528183}{3473230923504514068298735536222404533389561491910001729073776071337665411477940044334085} a^{12} - \frac{254870309987033690961918519582069615762118828108110180126647191780729704348163732545513}{3473230923504514068298735536222404533389561491910001729073776071337665411477940044334085} a^{11} - \frac{7137561232815642632833169865233721350954778126569888339513890333138911065335236038524152}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a^{10} + \frac{1464372133254192516700350262944406588973544513938978463185125897396391164518216067853771}{3473230923504514068298735536222404533389561491910001729073776071337665411477940044334085} a^{9} + \frac{554935401921677123662093284597593604369863647325367592713903532291610605399884041503999}{3473230923504514068298735536222404533389561491910001729073776071337665411477940044334085} a^{8} - \frac{3201982706855828327574813864165214906485868287959691106201510967497885343826277917356618}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a^{7} - \frac{7235948755321918604581278993498054421517123098863751891193471726390759767142808571663929}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a^{6} + \frac{765204094757503304220812114007682694591125117086027739425404715659623264410238262245746}{3473230923504514068298735536222404533389561491910001729073776071337665411477940044334085} a^{5} - \frac{5434451652395165206884776138646698768697703746067543760446173439095861677227264547570893}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a^{4} + \frac{3488883299983270855262962311372032471299447854746509938735227780203100232078746416845164}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a^{3} - \frac{2482444479120133781479787660310344227414800006358889235287389129953464129857519837643406}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a^{2} + \frac{2117607011570641279396430814995271392622096320431340005481248824055318863189157986183228}{24312616464531598478091148753556831733726930443370012103516432499363657880345580310338595} a + \frac{136044142640178462051797812242706653779541970688047382782864950607688037637746552095736}{4862523292906319695618229750711366346745386088674002420703286499872731576069116062067719}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 175585963226000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1959552 |
| The 168 conjugacy class representatives for t21n124 are not computed |
| Character table for t21n124 is not computed |
Intermediate fields
| 7.7.192100033.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 42 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $21$ | ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | R | $21$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ | $21$ | $21$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | $21$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.6.0.1 | $x^{6} + 3 x^{2} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 577 | Data not computed | ||||||
| 10697503699 | Data not computed | ||||||