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Properties

Label 20.12.2430952563...5625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{15}\cdot 6029^{5}$
Root discriminant $29.46$
Ramified primes $5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 19, -83, -22, 572, -724, -961, 2707, -162, -3543, 2115, 1963, -2111, -129, 829, -254, -108, 71, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 6*x^18 + 71*x^17 - 108*x^16 - 254*x^15 + 829*x^14 - 129*x^13 - 2111*x^12 + 1963*x^11 + 2115*x^10 - 3543*x^9 - 162*x^8 + 2707*x^7 - 961*x^6 - 724*x^5 + 572*x^4 - 22*x^3 - 83*x^2 + 19*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 6*x^18 + 71*x^17 - 108*x^16 - 254*x^15 + 829*x^14 - 129*x^13 - 2111*x^12 + 1963*x^11 + 2115*x^10 - 3543*x^9 - 162*x^8 + 2707*x^7 - 961*x^6 - 724*x^5 + 572*x^4 - 22*x^3 - 83*x^2 + 19*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 6 x^{18} + 71 x^{17} - 108 x^{16} - 254 x^{15} + 829 x^{14} - 129 x^{13} - 2111 x^{12} + 1963 x^{11} + 2115 x^{10} - 3543 x^{9} - 162 x^{8} + 2707 x^{7} - 961 x^{6} - 724 x^{5} + 572 x^{4} - 22 x^{3} - 83 x^{2} + 19 x - 1 \)

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:  \(243095256386686436431884765625=5^{15}\cdot 6029^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.46$
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:  $5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{16} - \frac{1}{10} a^{15} + \frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{2} a^{12} - \frac{1}{10} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{2} a^{7} - \frac{3}{10} a^{6} - \frac{3}{10} a^{5} - \frac{1}{5} a^{4} - \frac{3}{10} a^{3} - \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{10} a^{17} + \frac{1}{10} a^{15} + \frac{3}{10} a^{13} + \frac{2}{5} a^{12} - \frac{1}{10} a^{11} + \frac{1}{5} a^{10} + \frac{3}{10} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{3}{10} a^{3} - \frac{3}{10} a^{2} + \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{850} a^{18} + \frac{2}{85} a^{17} - \frac{7}{170} a^{16} + \frac{101}{850} a^{15} + \frac{78}{425} a^{14} + \frac{53}{425} a^{13} - \frac{381}{850} a^{12} - \frac{327}{850} a^{11} - \frac{11}{170} a^{10} + \frac{61}{850} a^{9} + \frac{62}{425} a^{8} - \frac{83}{425} a^{7} - \frac{56}{425} a^{6} - \frac{31}{425} a^{5} - \frac{133}{425} a^{4} + \frac{9}{34} a^{3} + \frac{133}{425} a^{2} + \frac{106}{425} a - \frac{339}{850}$, $\frac{1}{227260597343908501750} a^{19} - \frac{31242701849201786}{113630298671954250875} a^{18} - \frac{518475548915155608}{22726059734390850175} a^{17} - \frac{1423907488236631757}{113630298671954250875} a^{16} - \frac{1084635283516797043}{13368270431994617750} a^{15} + \frac{13846288436715556509}{227260597343908501750} a^{14} + \frac{44296772438028474986}{113630298671954250875} a^{13} + \frac{8286831822483385913}{22726059734390850175} a^{12} - \frac{12903724972013358253}{113630298671954250875} a^{11} - \frac{10746059266974362582}{113630298671954250875} a^{10} - \frac{9491467298396643254}{113630298671954250875} a^{9} - \frac{28727850046025001969}{227260597343908501750} a^{8} + \frac{629107771862004687}{1818084778751268014} a^{7} + \frac{16116392180606609426}{113630298671954250875} a^{6} + \frac{27281959952172802193}{227260597343908501750} a^{5} - \frac{51637401959296425133}{227260597343908501750} a^{4} + \frac{36595275660600401093}{113630298671954250875} a^{3} - \frac{364015137467149994}{1748158441106988475} a^{2} + \frac{80093427145339209857}{227260597343908501750} a + \frac{39127763437977805553}{227260597343908501750}$

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:  \( 33379361.6833 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
6029Data not computed