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Properties

Label 20.8.44181154111...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{16}\cdot 6029^{7}$
Root discriminant $76.25$
Ramified primes $5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-176155, 5616100, 19666195, -16163815, 977675, -9627805, -3713896, -1305690, -552589, 85897, 105428, 33873, 15220, 4102, -1971, 281, -327, 0, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 2*x^18 - 327*x^16 + 281*x^15 - 1971*x^14 + 4102*x^13 + 15220*x^12 + 33873*x^11 + 105428*x^10 + 85897*x^9 - 552589*x^8 - 1305690*x^7 - 3713896*x^6 - 9627805*x^5 + 977675*x^4 - 16163815*x^3 + 19666195*x^2 + 5616100*x - 176155)
 
gp: K = bnfinit(x^20 - x^19 + 2*x^18 - 327*x^16 + 281*x^15 - 1971*x^14 + 4102*x^13 + 15220*x^12 + 33873*x^11 + 105428*x^10 + 85897*x^9 - 552589*x^8 - 1305690*x^7 - 3713896*x^6 - 9627805*x^5 + 977675*x^4 - 16163815*x^3 + 19666195*x^2 + 5616100*x - 176155, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 2 x^{18} - 327 x^{16} + 281 x^{15} - 1971 x^{14} + 4102 x^{13} + 15220 x^{12} + 33873 x^{11} + 105428 x^{10} + 85897 x^{9} - 552589 x^{8} - 1305690 x^{7} - 3713896 x^{6} - 9627805 x^{5} + 977675 x^{4} - 16163815 x^{3} + 19666195 x^{2} + 5616100 x - 176155 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(44181154111269498973595933380126953125=5^{16}\cdot 6029^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.25$
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:  $5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{65} a^{16} - \frac{32}{65} a^{15} + \frac{18}{65} a^{14} - \frac{2}{65} a^{13} - \frac{32}{65} a^{12} + \frac{9}{65} a^{11} - \frac{19}{65} a^{10} + \frac{2}{5} a^{9} + \frac{1}{65} a^{8} - \frac{29}{65} a^{7} - \frac{2}{5} a^{6} + \frac{1}{13} a^{4} + \frac{5}{13} a^{3} - \frac{6}{13} a^{2} + \frac{2}{13} a - \frac{6}{13}$, $\frac{1}{65} a^{17} - \frac{31}{65} a^{15} - \frac{11}{65} a^{14} - \frac{31}{65} a^{13} + \frac{5}{13} a^{12} + \frac{9}{65} a^{11} + \frac{3}{65} a^{10} - \frac{12}{65} a^{9} + \frac{3}{65} a^{8} + \frac{21}{65} a^{7} + \frac{1}{5} a^{6} + \frac{1}{13} a^{5} - \frac{2}{13} a^{4} - \frac{2}{13} a^{3} + \frac{5}{13} a^{2} + \frac{6}{13} a + \frac{3}{13}$, $\frac{1}{65} a^{18} - \frac{28}{65} a^{15} + \frac{7}{65} a^{14} + \frac{28}{65} a^{13} - \frac{8}{65} a^{12} + \frac{22}{65} a^{11} - \frac{16}{65} a^{10} + \frac{29}{65} a^{9} - \frac{1}{5} a^{8} + \frac{24}{65} a^{7} - \frac{21}{65} a^{6} - \frac{2}{13} a^{5} + \frac{3}{13} a^{4} + \frac{4}{13} a^{3} + \frac{2}{13} a^{2} - \frac{4}{13}$, $\frac{1}{17171898824021089119349478429120244151781716227520021598966573453525} a^{19} + \frac{2301841508060359279838677163010078180501702066305627331754435}{686875952960843564773979137164809766071268649100800863958662938141} a^{18} + \frac{96760381099869941814636353418196213579101080684162402299225704357}{17171898824021089119349478429120244151781716227520021598966573453525} a^{17} - \frac{8943417681190132220332065879962809438401156051441051660154855264}{2453128403431584159907068347017177735968816603931431656995224779075} a^{16} + \frac{59153569809590790927020647433389401412889790921694984359492245568}{202022339106130460227640922695532284138608426206117901164312628865} a^{15} - \frac{7769343879568307784246759907991170017754979810577863242790732321634}{17171898824021089119349478429120244151781716227520021598966573453525} a^{14} - \frac{159488271687370038337349343118804752269357009605761528573546270749}{686875952960843564773979137164809766071268649100800863958662938141} a^{13} + \frac{209697143876419158916597166930473317490048186724862507948054863941}{2453128403431584159907068347017177735968816603931431656995224779075} a^{12} - \frac{273042581889876619119550900491576430608330306062449874178470339086}{1320915294155468393796113725316941857829362786732309353766659496425} a^{11} - \frac{2975815303961419194986453047390608272181516889663466181254022073}{28860334158018637175377274670790326305515489458016843023473232695} a^{10} - \frac{6953213848835026939198247026181236207394004923094455997304672063247}{17171898824021089119349478429120244151781716227520021598966573453525} a^{9} - \frac{162152665338917600249114691681339285011068232255034752139744651109}{490625680686316831981413669403435547193763320786286331399044955815} a^{8} + \frac{5384758512563139581847545856701605053895811524512664437008930607696}{17171898824021089119349478429120244151781716227520021598966573453525} a^{7} + \frac{5426233511717120930954723443963302826048695003460932066542300980301}{17171898824021089119349478429120244151781716227520021598966573453525} a^{6} - \frac{805952636620405498391824600224051538940369723404669452452070760569}{3434379764804217823869895685824048830356343245504004319793314690705} a^{5} - \frac{290701526470541151701941922745302103911624188587879357447651063548}{686875952960843564773979137164809766071268649100800863958662938141} a^{4} - \frac{108376634086899102902648471323207957100106940958313647148041115539}{686875952960843564773979137164809766071268649100800863958662938141} a^{3} + \frac{1689532984027277489963733263233949183739530187362651460727407217307}{3434379764804217823869895685824048830356343245504004319793314690705} a^{2} + \frac{929457477602096539007004215912735068680285649108571005252008426826}{3434379764804217823869895685824048830356343245504004319793314690705} a - \frac{12097554687842929185261538739844136191872624541805967573834524969}{37740436975870525537031820723341195937981793906637410107618842755}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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:  \( 235394886379 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 40 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 $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
6029Data not computed