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Properties

Label 20.4.34919910159...6481.2
Degree $20$
Signature $[4, 8]$
Discriminant $3^{4}\cdot 401^{11}$
Root discriminant $33.66$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times C_2^4:C_5).C_2$ (as 20T84)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![747, -3624, 6295, -388, -9436, 6499, 7412, -8197, -5114, 6966, 2253, -2226, -2844, 1887, 658, -586, -37, 85, -10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 10*x^18 + 85*x^17 - 37*x^16 - 586*x^15 + 658*x^14 + 1887*x^13 - 2844*x^12 - 2226*x^11 + 2253*x^10 + 6966*x^9 - 5114*x^8 - 8197*x^7 + 7412*x^6 + 6499*x^5 - 9436*x^4 - 388*x^3 + 6295*x^2 - 3624*x + 747)
 
gp: K = bnfinit(x^20 - 4*x^19 - 10*x^18 + 85*x^17 - 37*x^16 - 586*x^15 + 658*x^14 + 1887*x^13 - 2844*x^12 - 2226*x^11 + 2253*x^10 + 6966*x^9 - 5114*x^8 - 8197*x^7 + 7412*x^6 + 6499*x^5 - 9436*x^4 - 388*x^3 + 6295*x^2 - 3624*x + 747, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 10 x^{18} + 85 x^{17} - 37 x^{16} - 586 x^{15} + 658 x^{14} + 1887 x^{13} - 2844 x^{12} - 2226 x^{11} + 2253 x^{10} + 6966 x^{9} - 5114 x^{8} - 8197 x^{7} + 7412 x^{6} + 6499 x^{5} - 9436 x^{4} - 388 x^{3} + 6295 x^{2} - 3624 x + 747 \)

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:  \(3491991015954496398424073156481=3^{4}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.66$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{4947} a^{18} + \frac{812}{4947} a^{17} + \frac{53}{4947} a^{16} - \frac{31}{4947} a^{15} + \frac{823}{1649} a^{14} - \frac{856}{4947} a^{13} + \frac{717}{1649} a^{12} + \frac{554}{1649} a^{11} + \frac{1718}{4947} a^{10} + \frac{1160}{4947} a^{9} - \frac{95}{1649} a^{8} - \frac{127}{1649} a^{7} - \frac{137}{291} a^{6} + \frac{702}{1649} a^{5} + \frac{46}{291} a^{4} + \frac{287}{4947} a^{3} - \frac{1174}{4947} a^{2} + \frac{64}{4947} a - \frac{671}{1649}$, $\frac{1}{5122900212047879819093285105089099689} a^{19} + \frac{300350105510141648733128241709102}{5122900212047879819093285105089099689} a^{18} - \frac{32204619363469145998650185835980657}{301347071296934107005487359122888217} a^{17} + \frac{4827681280342172407572765200959972}{5122900212047879819093285105089099689} a^{16} + \frac{2828205067525020098307886528573}{1551923723734589463524169980335989} a^{15} - \frac{405987414711140342312230101709380580}{5122900212047879819093285105089099689} a^{14} + \frac{800006683937253441042881872435275220}{5122900212047879819093285105089099689} a^{13} + \frac{2528923159831467047318818811646175724}{5122900212047879819093285105089099689} a^{12} + \frac{1838833284676119494008547191888183121}{5122900212047879819093285105089099689} a^{11} + \frac{2450452456699537769518879434242485381}{5122900212047879819093285105089099689} a^{10} - \frac{678062573887728505635072208903286746}{1707633404015959939697761701696366563} a^{9} - \frac{1696721033239713168128855153319041780}{5122900212047879819093285105089099689} a^{8} - \frac{1114578761714604989367054774676258576}{5122900212047879819093285105089099689} a^{7} - \frac{60172490074512276706612711087790890}{1707633404015959939697761701696366563} a^{6} + \frac{2455554716858833874773053846227814374}{5122900212047879819093285105089099689} a^{5} - \frac{76464838754117674045059786009676359}{1707633404015959939697761701696366563} a^{4} + \frac{261759421887689491559466911169179699}{1707633404015959939697761701696366563} a^{3} + \frac{210615478902809613130678545690984382}{1707633404015959939697761701696366563} a^{2} - \frac{1821331505555880822386768217041933273}{5122900212047879819093285105089099689} a - \frac{360658670921941906697004531894696817}{1707633404015959939697761701696366563}$

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:  $11$
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:  \( 23219613.5506 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_2^4:C_5).C_2$ (as 20T84):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $(C_2\times C_2^4:C_5).C_2$
Character table for $(C_2\times C_2^4:C_5).C_2$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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

Sibling fields

Degree 20 siblings: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed