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Properties

Label 20.0.69189328977...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{15}\cdot 61^{5}$
Root discriminant $24.66$
Ramified primes $2, 5, 61$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $D_5^2:C_4$ (as 20T94)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![841, -580, -4584, 5688, 7107, -15200, 2512, 13508, -10644, 612, 3116, -2584, 2100, -1060, 252, -48, 51, -12, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 4*x^18 - 12*x^17 + 51*x^16 - 48*x^15 + 252*x^14 - 1060*x^13 + 2100*x^12 - 2584*x^11 + 3116*x^10 + 612*x^9 - 10644*x^8 + 13508*x^7 + 2512*x^6 - 15200*x^5 + 7107*x^4 + 5688*x^3 - 4584*x^2 - 580*x + 841)
 
gp: K = bnfinit(x^20 - 4*x^19 + 4*x^18 - 12*x^17 + 51*x^16 - 48*x^15 + 252*x^14 - 1060*x^13 + 2100*x^12 - 2584*x^11 + 3116*x^10 + 612*x^9 - 10644*x^8 + 13508*x^7 + 2512*x^6 - 15200*x^5 + 7107*x^4 + 5688*x^3 - 4584*x^2 - 580*x + 841, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 4 x^{18} - 12 x^{17} + 51 x^{16} - 48 x^{15} + 252 x^{14} - 1060 x^{13} + 2100 x^{12} - 2584 x^{11} + 3116 x^{10} + 612 x^{9} - 10644 x^{8} + 13508 x^{7} + 2512 x^{6} - 15200 x^{5} + 7107 x^{4} + 5688 x^{3} - 4584 x^{2} - 580 x + 841 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6918932897792000000000000000=2^{28}\cdot 5^{15}\cdot 61^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.66$
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:  $2, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{2} a^{8} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{47941489809307238876366487371733908396} a^{19} - \frac{698736210979036305762241079500282203}{23970744904653619438183243685866954198} a^{18} + \frac{4227517922172670795701212701262159}{42463675650404994576055347539179724} a^{17} - \frac{2908486848617120914488667273071505780}{11985372452326809719091621842933477099} a^{16} + \frac{4706703683184245969652394065916763015}{23970744904653619438183243685866954198} a^{15} - \frac{447371911969892120756178895432673767}{23970744904653619438183243685866954198} a^{14} + \frac{2581258383408263493124054863060530144}{11985372452326809719091621842933477099} a^{13} + \frac{2387104768898963430590243705865947040}{11985372452326809719091621842933477099} a^{12} + \frac{1606569187881287091924429104853584805}{23970744904653619438183243685866954198} a^{11} - \frac{694051024163703040537489475431779072}{11985372452326809719091621842933477099} a^{10} - \frac{5041666502127290691607667130353282319}{23970744904653619438183243685866954198} a^{9} - \frac{1561294408009767143208158831157156151}{23970744904653619438183243685866954198} a^{8} + \frac{7487286842809816842302999963159529603}{23970744904653619438183243685866954198} a^{7} - \frac{8022426864483473500381369627799779995}{23970744904653619438183243685866954198} a^{6} - \frac{1688143313335433786643149198241314043}{11985372452326809719091621842933477099} a^{5} + \frac{5398146829573501974490259178992582241}{11985372452326809719091621842933477099} a^{4} - \frac{20732716183891544652811144526703922555}{47941489809307238876366487371733908396} a^{3} + \frac{2461890527715418875396593010137116877}{23970744904653619438183243685866954198} a^{2} + \frac{14664944488148778862542712553873737715}{47941489809307238876366487371733908396} a + \frac{97775917850761648936729299276517963}{826577410505297222006318747788515662}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $9$
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:  \( 66409.6997786 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2:C_4$ (as 20T94):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $D_5^2:C_4$
Character table for $D_5^2:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.122000.2, 10.2.2976800000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
5Data not computed
$61$61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$