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Properties

Label 20.4.21333423461...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{15}\cdot 31^{18}$
Root discriminant $73.53$
Ramified primes $5, 31$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-968125, -3537500, -3259625, 1354000, 3604650, 1275150, -1307700, -986175, 94360, 232525, 1825, -37645, -3389, 7124, -502, -485, -89, 85, 2, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 2*x^18 + 85*x^17 - 89*x^16 - 485*x^15 - 502*x^14 + 7124*x^13 - 3389*x^12 - 37645*x^11 + 1825*x^10 + 232525*x^9 + 94360*x^8 - 986175*x^7 - 1307700*x^6 + 1275150*x^5 + 3604650*x^4 + 1354000*x^3 - 3259625*x^2 - 3537500*x - 968125)
 
gp: K = bnfinit(x^20 - 7*x^19 + 2*x^18 + 85*x^17 - 89*x^16 - 485*x^15 - 502*x^14 + 7124*x^13 - 3389*x^12 - 37645*x^11 + 1825*x^10 + 232525*x^9 + 94360*x^8 - 986175*x^7 - 1307700*x^6 + 1275150*x^5 + 3604650*x^4 + 1354000*x^3 - 3259625*x^2 - 3537500*x - 968125, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 2 x^{18} + 85 x^{17} - 89 x^{16} - 485 x^{15} - 502 x^{14} + 7124 x^{13} - 3389 x^{12} - 37645 x^{11} + 1825 x^{10} + 232525 x^{9} + 94360 x^{8} - 986175 x^{7} - 1307700 x^{6} + 1275150 x^{5} + 3604650 x^{4} + 1354000 x^{3} - 3259625 x^{2} - 3537500 x - 968125 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21333423461884919389012763702392578125=5^{15}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.53$
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:  $5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{30} a^{12} - \frac{7}{30} a^{11} - \frac{13}{30} a^{10} - \frac{1}{2} a^{9} - \frac{7}{15} a^{8} - \frac{1}{2} a^{7} + \frac{4}{15} a^{6} + \frac{3}{10} a^{5} + \frac{11}{30} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{30} a^{13} - \frac{1}{15} a^{11} + \frac{7}{15} a^{10} + \frac{1}{30} a^{9} + \frac{7}{30} a^{8} - \frac{7}{30} a^{7} + \frac{1}{6} a^{6} + \frac{7}{15} a^{5} - \frac{13}{30} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{60} a^{14} - \frac{1}{60} a^{13} - \frac{1}{60} a^{12} - \frac{1}{10} a^{11} - \frac{11}{60} a^{10} - \frac{2}{5} a^{9} - \frac{13}{60} a^{8} - \frac{1}{20} a^{7} - \frac{7}{15} a^{6} + \frac{1}{5} a^{5} - \frac{1}{60} a^{4} - \frac{1}{6} a^{2} - \frac{1}{4}$, $\frac{1}{42600} a^{15} - \frac{43}{10650} a^{14} - \frac{29}{3550} a^{13} + \frac{1}{2840} a^{12} + \frac{5171}{42600} a^{11} + \frac{997}{8520} a^{10} + \frac{1011}{14200} a^{9} + \frac{263}{5325} a^{8} - \frac{10319}{42600} a^{7} + \frac{368}{1065} a^{6} - \frac{2719}{8520} a^{5} + \frac{1441}{8520} a^{4} - \frac{751}{2130} a^{3} - \frac{65}{213} a^{2} + \frac{77}{1704} a - \frac{625}{1704}$, $\frac{1}{1022400} a^{16} + \frac{1}{204480} a^{15} - \frac{299}{127800} a^{14} - \frac{7621}{1022400} a^{13} + \frac{3203}{511200} a^{12} - \frac{2107}{255600} a^{11} - \frac{44137}{170400} a^{10} + \frac{39061}{204480} a^{9} + \frac{298189}{1022400} a^{8} + \frac{444637}{1022400} a^{7} + \frac{9853}{22720} a^{6} - \frac{8137}{20448} a^{5} - \frac{4969}{68160} a^{4} + \frac{7939}{25560} a^{3} + \frac{18229}{40896} a^{2} - \frac{277}{1278} a + \frac{1555}{40896}$, $\frac{1}{2044800} a^{17} - \frac{17}{2044800} a^{15} + \frac{499}{2044800} a^{14} + \frac{9077}{681600} a^{13} - \frac{743}{340800} a^{12} + \frac{2309}{14400} a^{11} + \frac{79339}{408960} a^{10} + \frac{599}{21300} a^{9} - \frac{123817}{511200} a^{8} - \frac{14411}{51120} a^{7} + \frac{40393}{81792} a^{6} + \frac{41639}{408960} a^{5} + \frac{202079}{408960} a^{4} + \frac{25949}{81792} a^{3} + \frac{1559}{81792} a^{2} + \frac{26195}{81792} a + \frac{5377}{81792}$, $\frac{1}{20448000} a^{18} + \frac{1}{6816000} a^{17} + \frac{7}{20448000} a^{16} - \frac{1}{102240} a^{15} - \frac{457}{284000} a^{14} - \frac{953}{454400} a^{13} + \frac{31531}{2556000} a^{12} + \frac{4159049}{20448000} a^{11} - \frac{523411}{2272000} a^{10} + \frac{1753}{40896} a^{9} + \frac{330689}{1022400} a^{8} + \frac{249149}{4089600} a^{7} - \frac{110669}{2044800} a^{6} - \frac{18415}{40896} a^{5} - \frac{76937}{408960} a^{4} + \frac{4139}{81792} a^{3} + \frac{10781}{51120} a^{2} + \frac{35737}{81792} a + \frac{23749}{54528}$, $\frac{1}{90325637578105648617466240439616000} a^{19} - \frac{536119467430049117371795873}{45162818789052824308733120219808000} a^{18} - \frac{1901616796367705550499079}{301085458593685495391554134798720} a^{17} - \frac{3449011323442327732190884121}{30108545859368549539155413479872000} a^{16} + \frac{156948070022608961057373277739}{22581409394526412154366560109904000} a^{15} - \frac{288001875243216995977014328780769}{90325637578105648617466240439616000} a^{14} - \frac{636372012594087625294034498193347}{90325637578105648617466240439616000} a^{13} - \frac{49789200796225973726978790732343}{90325637578105648617466240439616000} a^{12} + \frac{34750675407723279736623099741227}{564535234863160303859164002747600} a^{11} + \frac{4422055621077281987444821575553599}{10036181953122849846385137826624000} a^{10} + \frac{525396604925158809595220096348677}{1129070469726320607718328005495200} a^{9} + \frac{4336223851600415989651166135023097}{18065127515621129723493248087923200} a^{8} - \frac{8149677562706932827357285377871179}{18065127515621129723493248087923200} a^{7} - \frac{991332788752440164018810209739579}{9032563757810564861746624043961600} a^{6} + \frac{56088168333616975615360798048093}{120434183437474198156621653919488} a^{5} - \frac{23968702771850621977676928294653}{301085458593685495391554134798720} a^{4} - \frac{230456656482631312584576715923299}{602170917187370990783108269597440} a^{3} - \frac{65906165739267015084229696452917}{1806512751562112972349324808792320} a^{2} + \frac{72065608165817269172243302265035}{240868366874948396313243307838976} a - \frac{32481164315744582797495503247471}{80289455624982798771081102612992}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $11$
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:  \( 46639726240.276184 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.120125.1, 5.1.115440125.1, 10.2.66632112300078125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$31$31.10.9.8$x^{10} + 521017$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.9.8$x^{10} + 521017$$10$$1$$9$$C_{10}$$[\ ]_{10}$