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Properties

Label 20.4.10237178602...0000.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{15}\cdot 13^{15}$
Root discriminant $39.86$
Ramified primes $2, 5, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1036, 736, 3180, 11392, -10440, 37406, -40484, 37180, -35540, 13358, -11727, 6169, 435, 210, -41, -277, 95, -26, 13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 13*x^18 - 26*x^17 + 95*x^16 - 277*x^15 - 41*x^14 + 210*x^13 + 435*x^12 + 6169*x^11 - 11727*x^10 + 13358*x^9 - 35540*x^8 + 37180*x^7 - 40484*x^6 + 37406*x^5 - 10440*x^4 + 11392*x^3 + 3180*x^2 + 736*x + 1036)
 
gp: K = bnfinit(x^20 - 5*x^19 + 13*x^18 - 26*x^17 + 95*x^16 - 277*x^15 - 41*x^14 + 210*x^13 + 435*x^12 + 6169*x^11 - 11727*x^10 + 13358*x^9 - 35540*x^8 + 37180*x^7 - 40484*x^6 + 37406*x^5 - 10440*x^4 + 11392*x^3 + 3180*x^2 + 736*x + 1036, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 13 x^{18} - 26 x^{17} + 95 x^{16} - 277 x^{15} - 41 x^{14} + 210 x^{13} + 435 x^{12} + 6169 x^{11} - 11727 x^{10} + 13358 x^{9} - 35540 x^{8} + 37180 x^{7} - 40484 x^{6} + 37406 x^{5} - 10440 x^{4} + 11392 x^{3} + 3180 x^{2} + 736 x + 1036 \)

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:  \(102371786028181514000000000000000=2^{16}\cdot 5^{15}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.86$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{14} + \frac{1}{6} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{15} + \frac{1}{6} a^{14} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{3} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{18} + \frac{1}{6} a^{15} - \frac{1}{6} a^{14} - \frac{1}{6} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{1244062596247638775476798325570489217875386} a^{19} + \frac{441462238170103115025016988045059151330}{622031298123819387738399162785244608937693} a^{18} + \frac{43182684324072914817728258212153405894103}{1244062596247638775476798325570489217875386} a^{17} - \frac{7081827538168961666659598749120211343093}{207343766041273129246133054261748202979231} a^{16} - \frac{103922332845487487530584156028989443500970}{622031298123819387738399162785244608937693} a^{15} - \frac{57974548446255488429050397983459726619191}{1244062596247638775476798325570489217875386} a^{14} - \frac{65451692639407275863120775583406625006037}{1244062596247638775476798325570489217875386} a^{13} - \frac{143720604556091499144242102622661259272553}{622031298123819387738399162785244608937693} a^{12} + \frac{87289447639063863288609800292772108764230}{622031298123819387738399162785244608937693} a^{11} - \frac{297162815223155861772895350718833684476651}{1244062596247638775476798325570489217875386} a^{10} + \frac{15321710667267062116581248070895829706867}{414687532082546258492266108523496405958462} a^{9} - \frac{177272392722399960646898526582874841726849}{622031298123819387738399162785244608937693} a^{8} + \frac{342434171348832587281287259419050856205595}{1244062596247638775476798325570489217875386} a^{7} + \frac{406793644146068003208412599309751679183389}{1244062596247638775476798325570489217875386} a^{6} + \frac{63126656628315164976378511313646298066585}{207343766041273129246133054261748202979231} a^{5} - \frac{152008209289453792008939191179989526786805}{622031298123819387738399162785244608937693} a^{4} - \frac{182426467581787628754004716815532271661338}{622031298123819387738399162785244608937693} a^{3} + \frac{122154749675195675409892019869549846710124}{622031298123819387738399162785244608937693} a^{2} - \frac{38974387153380623856078393836377131598314}{207343766041273129246133054261748202979231} a - \frac{106450518979225113641758050239815050383383}{622031298123819387738399162785244608937693}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 175565790.70089695 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{65}) \), 4.4.274625.2, 5.1.338000.1, 10.2.7425860000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ $20$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
13Data not computed