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Properties

Label 20.12.9495293556...0625.2
Degree $20$
Signature $[12, 4]$
Discriminant $5^{14}\cdot 11^{5}\cdot 9931^{5}$
Root discriminant $56.09$
Ramified primes $5, 11, 9931$
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![-19, -479, -248, 3889, 3449, -12240, -17130, 14058, 36701, 8761, -23680, -15567, 4271, 5767, 217, -710, -5, 34, -17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 17*x^18 + 34*x^17 - 5*x^16 - 710*x^15 + 217*x^14 + 5767*x^13 + 4271*x^12 - 15567*x^11 - 23680*x^10 + 8761*x^9 + 36701*x^8 + 14058*x^7 - 17130*x^6 - 12240*x^5 + 3449*x^4 + 3889*x^3 - 248*x^2 - 479*x - 19)
 
gp: K = bnfinit(x^20 - 2*x^19 - 17*x^18 + 34*x^17 - 5*x^16 - 710*x^15 + 217*x^14 + 5767*x^13 + 4271*x^12 - 15567*x^11 - 23680*x^10 + 8761*x^9 + 36701*x^8 + 14058*x^7 - 17130*x^6 - 12240*x^5 + 3449*x^4 + 3889*x^3 - 248*x^2 - 479*x - 19, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 17 x^{18} + 34 x^{17} - 5 x^{16} - 710 x^{15} + 217 x^{14} + 5767 x^{13} + 4271 x^{12} - 15567 x^{11} - 23680 x^{10} + 8761 x^{9} + 36701 x^{8} + 14058 x^{7} - 17130 x^{6} - 12240 x^{5} + 3449 x^{4} + 3889 x^{3} - 248 x^{2} - 479 x - 19 \)

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:  \(94952935569355941136243902587890625=5^{14}\cdot 11^{5}\cdot 9931^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.09$
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:  $5, 11, 9931$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{11499298573629617527378072809043813} a^{19} - \frac{46343962761346640535490123157932}{11499298573629617527378072809043813} a^{18} - \frac{1600654740887837336929020196410476}{11499298573629617527378072809043813} a^{17} + \frac{3022602951823567607344587127773791}{11499298573629617527378072809043813} a^{16} - \frac{1517111509359779649598497230132649}{11499298573629617527378072809043813} a^{15} - \frac{4953191498189944929693828772945036}{11499298573629617527378072809043813} a^{14} - \frac{5237045027565772534995815479375796}{11499298573629617527378072809043813} a^{13} + \frac{2351446337332866506846354221098496}{11499298573629617527378072809043813} a^{12} - \frac{4764936551783428046596443355129735}{11499298573629617527378072809043813} a^{11} + \frac{3683848484606268587595293416386004}{11499298573629617527378072809043813} a^{10} + \frac{4397657466269232241156587580432117}{11499298573629617527378072809043813} a^{9} + \frac{2046117156837664401376893839711665}{11499298573629617527378072809043813} a^{8} + \frac{2265251155073550053098528732563927}{11499298573629617527378072809043813} a^{7} + \frac{3376193599377014463692912399259852}{11499298573629617527378072809043813} a^{6} + \frac{1563051356962604268685609306100459}{11499298573629617527378072809043813} a^{5} + \frac{2014705350350859044073489192143930}{11499298573629617527378072809043813} a^{4} + \frac{5136696302288300663850681676635731}{11499298573629617527378072809043813} a^{3} + \frac{1596954854287208687541955294554297}{11499298573629617527378072809043813} a^{2} + \frac{5313620741740159201821223451139483}{11499298573629617527378072809043813} a - \frac{94961821011292084903256886289689}{11499298573629617527378072809043813}$

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:  \( 23519527421.7 \) (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

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R $20$ R $20$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
9931Data not computed