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Properties

Label 20.16.3102574545...0000.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{20}\cdot 5^{10}\cdot 13^{6}\cdot 29^{4}\cdot 31^{6}$
Root discriminant $53.04$
Ramified primes $2, 5, 13, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -60, 132, 1744, -9800, 16948, 7896, -49768, 17388, 49200, -25750, -24492, 13616, 7042, -3642, -1202, 529, 109, -38, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 38*x^18 + 109*x^17 + 529*x^16 - 1202*x^15 - 3642*x^14 + 7042*x^13 + 13616*x^12 - 24492*x^11 - 25750*x^10 + 49200*x^9 + 17388*x^8 - 49768*x^7 + 7896*x^6 + 16948*x^5 - 9800*x^4 + 1744*x^3 + 132*x^2 - 60*x - 4)
 
gp: K = bnfinit(x^20 - 4*x^19 - 38*x^18 + 109*x^17 + 529*x^16 - 1202*x^15 - 3642*x^14 + 7042*x^13 + 13616*x^12 - 24492*x^11 - 25750*x^10 + 49200*x^9 + 17388*x^8 - 49768*x^7 + 7896*x^6 + 16948*x^5 - 9800*x^4 + 1744*x^3 + 132*x^2 - 60*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 38 x^{18} + 109 x^{17} + 529 x^{16} - 1202 x^{15} - 3642 x^{14} + 7042 x^{13} + 13616 x^{12} - 24492 x^{11} - 25750 x^{10} + 49200 x^{9} + 17388 x^{8} - 49768 x^{7} + 7896 x^{6} + 16948 x^{5} - 9800 x^{4} + 1744 x^{3} + 132 x^{2} - 60 x - 4 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31025745455060872059381760000000000=2^{20}\cdot 5^{10}\cdot 13^{6}\cdot 29^{4}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.04$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{140434061108229884795923493588} a^{19} + \frac{4501880761844844487455348305}{70217030554114942397961746794} a^{18} + \frac{16786790955530330607471215023}{70217030554114942397961746794} a^{17} - \frac{649493931879146906421651915}{140434061108229884795923493588} a^{16} + \frac{29763793412883847174087412563}{140434061108229884795923493588} a^{15} + \frac{639430496731916236819754942}{35108515277057471198980873397} a^{14} + \frac{5210273408552400554164983689}{70217030554114942397961746794} a^{13} + \frac{5950097879913845684145889597}{35108515277057471198980873397} a^{12} - \frac{32423000555878774252640789131}{70217030554114942397961746794} a^{11} + \frac{11186165720354616378015594135}{35108515277057471198980873397} a^{10} - \frac{9805466981700331300175473415}{35108515277057471198980873397} a^{9} + \frac{15704027584345774746248051578}{35108515277057471198980873397} a^{8} - \frac{18824068564119611263035334983}{70217030554114942397961746794} a^{7} + \frac{10279226312436694870643781201}{35108515277057471198980873397} a^{6} - \frac{3533811503223110815603693138}{35108515277057471198980873397} a^{5} + \frac{5048341667501240210908308111}{35108515277057471198980873397} a^{4} + \frac{11534638805619636426549124735}{35108515277057471198980873397} a^{3} + \frac{1390191317309671756160537641}{35108515277057471198980873397} a^{2} - \frac{15910920884393716530565688293}{35108515277057471198980873397} a - \frac{11930436396123734597607330440}{35108515277057471198980873397}$

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:  $17$
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:  \( 45070528663.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
5Data not computed
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.6.5.3$x^{6} - 74431$$6$$1$$5$$C_6$$[\ ]_{6}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$