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Properties

Label 20.12.2572121384...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $37.20$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89, 482, -2748, -12394, 4549, 38316, -5382, -42378, -3490, 33924, -38, -13418, -154, 3420, 74, -470, -111, 86, 6, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 6*x^18 + 86*x^17 - 111*x^16 - 470*x^15 + 74*x^14 + 3420*x^13 - 154*x^12 - 13418*x^11 - 38*x^10 + 33924*x^9 - 3490*x^8 - 42378*x^7 - 5382*x^6 + 38316*x^5 + 4549*x^4 - 12394*x^3 - 2748*x^2 + 482*x + 89)
 
gp: K = bnfinit(x^20 - 8*x^19 + 6*x^18 + 86*x^17 - 111*x^16 - 470*x^15 + 74*x^14 + 3420*x^13 - 154*x^12 - 13418*x^11 - 38*x^10 + 33924*x^9 - 3490*x^8 - 42378*x^7 - 5382*x^6 + 38316*x^5 + 4549*x^4 - 12394*x^3 - 2748*x^2 + 482*x + 89, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 6 x^{18} + 86 x^{17} - 111 x^{16} - 470 x^{15} + 74 x^{14} + 3420 x^{13} - 154 x^{12} - 13418 x^{11} - 38 x^{10} + 33924 x^{9} - 3490 x^{8} - 42378 x^{7} - 5382 x^{6} + 38316 x^{5} + 4549 x^{4} - 12394 x^{3} - 2748 x^{2} + 482 x + 89 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(25721213848857003468800000000000=2^{20}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.20$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{20} a^{18} - \frac{1}{10} a^{17} - \frac{1}{4} a^{16} + \frac{1}{5} a^{15} - \frac{1}{10} a^{14} + \frac{1}{10} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{10} a^{8} + \frac{1}{10} a^{6} - \frac{3}{10} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{7}{20} a^{2} + \frac{1}{10} a - \frac{9}{20}$, $\frac{1}{33560483526132395779721944181020} a^{19} - \frac{271760465119489948019045629443}{16780241763066197889860972090510} a^{18} - \frac{3620922591144594727934358362777}{33560483526132395779721944181020} a^{17} + \frac{2041054290341284126604463382721}{8390120881533098944930486045255} a^{16} + \frac{405222299773882327721406035541}{16780241763066197889860972090510} a^{15} - \frac{112270465907879734867776308549}{1678024176306619788986097209051} a^{14} - \frac{2306913584856122170697843062417}{16780241763066197889860972090510} a^{13} - \frac{726836401026958959104054462607}{3356048352613239577972194418102} a^{12} - \frac{2670116537984787399245249064799}{16780241763066197889860972090510} a^{11} + \frac{144015800153635030893865445016}{1678024176306619788986097209051} a^{10} - \frac{7618883035092572745967858389067}{16780241763066197889860972090510} a^{9} - \frac{803306128621099194322070491097}{8390120881533098944930486045255} a^{8} - \frac{5530541559987792342585948219309}{16780241763066197889860972090510} a^{7} + \frac{2022563052160278051331952562754}{8390120881533098944930486045255} a^{6} + \frac{4847238928859366282346482659879}{16780241763066197889860972090510} a^{5} - \frac{5897054746682096999245384858081}{16780241763066197889860972090510} a^{4} + \frac{1777527020867859963826021653361}{33560483526132395779721944181020} a^{3} - \frac{5598294501737493245442329972933}{16780241763066197889860972090510} a^{2} - \frac{16260014596107901113807534516217}{33560483526132395779721944181020} a - \frac{1606852313137777659220761301731}{8390120881533098944930486045255}$

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:  $15$
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:  \( 364656521.981 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed