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Properties

Label 20.10.1002402935...3264.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 3^{19}\cdot 13^{7}\cdot 107^{4}$
Root discriminant $35.49$
Ramified primes $2, 3, 13, 107$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -12, 48, 306, 498, -210, -1782, -1302, 637, 156, -315, 190, -300, -330, 81, 30, -15, 22, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 3*x^18 + 22*x^17 - 15*x^16 + 30*x^15 + 81*x^14 - 330*x^13 - 300*x^12 + 190*x^11 - 315*x^10 + 156*x^9 + 637*x^8 - 1302*x^7 - 1782*x^6 - 210*x^5 + 498*x^4 + 306*x^3 + 48*x^2 - 12*x - 3)
 
gp: K = bnfinit(x^20 - 6*x^19 + 3*x^18 + 22*x^17 - 15*x^16 + 30*x^15 + 81*x^14 - 330*x^13 - 300*x^12 + 190*x^11 - 315*x^10 + 156*x^9 + 637*x^8 - 1302*x^7 - 1782*x^6 - 210*x^5 + 498*x^4 + 306*x^3 + 48*x^2 - 12*x - 3, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 3 x^{18} + 22 x^{17} - 15 x^{16} + 30 x^{15} + 81 x^{14} - 330 x^{13} - 300 x^{12} + 190 x^{11} - 315 x^{10} + 156 x^{9} + 637 x^{8} - 1302 x^{7} - 1782 x^{6} - 210 x^{5} + 498 x^{4} + 306 x^{3} + 48 x^{2} - 12 x - 3 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-10024029355890679518948322443264=-\,2^{20}\cdot 3^{19}\cdot 13^{7}\cdot 107^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.49$
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:  $2, 3, 13, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} + \frac{1}{3} a^{9}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} + \frac{1}{3} a^{10}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{8}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{17} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{4}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{5142540516429754795590639} a^{19} + \frac{42922625028678435352147}{1714180172143251598530213} a^{18} - \frac{407927551532775624769267}{5142540516429754795590639} a^{17} + \frac{101258446751242531947919}{1714180172143251598530213} a^{16} + \frac{244883726385679480330256}{1714180172143251598530213} a^{15} + \frac{19388823673217255317582}{244882881734750228361459} a^{14} + \frac{248521809483340002568891}{571393390714417199510071} a^{13} + \frac{55353556317474030133523}{571393390714417199510071} a^{12} + \frac{307471707182708787641719}{1714180172143251598530213} a^{11} - \frac{574515326808131768570996}{5142540516429754795590639} a^{10} - \frac{23578984828691129494135}{1714180172143251598530213} a^{9} - \frac{185913257567650832025007}{734648645204250685084377} a^{8} + \frac{17272229699212865347919}{244882881734750228361459} a^{7} + \frac{13442057559788077982437}{81627627244916742787153} a^{6} - \frac{49303272793574580016567}{571393390714417199510071} a^{5} + \frac{832928133618347277338906}{1714180172143251598530213} a^{4} - \frac{475938589424748714923072}{1714180172143251598530213} a^{3} + \frac{442441286375412141540691}{1714180172143251598530213} a^{2} + \frac{274076744895660589226440}{1714180172143251598530213} a + \frac{49898830131102724727740}{1714180172143251598530213}$

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:  $14$
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:  \( 392839564.457 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.6.38998285028352.1

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

Sibling fields

Degree 20 sibling: 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R $20$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ $20$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.6.6.7$x^{6} + 3 x + 6$$6$$1$$6$$C_3^2:D_4$$[5/4, 5/4]_{4}^{2}$
3.6.6.7$x^{6} + 3 x + 6$$6$$1$$6$$C_3^2:D_4$$[5/4, 5/4]_{4}^{2}$
3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
107Data not computed