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Properties

Label 20.12.2549038515...0000.5
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{15}\cdot 6029^{5}$
Root discriminant $58.93$
Ramified primes $2, 5, 6029$
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![753625, 0, -5275375, 0, 2937550, 0, 1049700, 0, -289765, 0, -57135, 0, 10599, 0, 1169, 0, -170, 0, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^18 - 170*x^16 + 1169*x^14 + 10599*x^12 - 57135*x^10 - 289765*x^8 + 1049700*x^6 + 2937550*x^4 - 5275375*x^2 + 753625)
 
gp: K = bnfinit(x^20 - 8*x^18 - 170*x^16 + 1169*x^14 + 10599*x^12 - 57135*x^10 - 289765*x^8 + 1049700*x^6 + 2937550*x^4 - 5275375*x^2 + 753625, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{18} - 170 x^{16} + 1169 x^{14} + 10599 x^{12} - 57135 x^{10} - 289765 x^{8} + 1049700 x^{6} + 2937550 x^{4} - 5275375 x^{2} + 753625 \)

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:  \(254903851560926116768000000000000000=2^{20}\cdot 5^{15}\cdot 6029^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.93$
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:  $2, 5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{10} - \frac{1}{5} a^{6} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{1}{5} a^{7} - \frac{1}{5} a^{5}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{11} - \frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{2}{5} a^{5}$, $\frac{1}{425} a^{16} - \frac{23}{425} a^{14} - \frac{7}{85} a^{12} + \frac{149}{425} a^{10} - \frac{11}{425} a^{8} + \frac{28}{85} a^{6} - \frac{26}{85} a^{4} + \frac{5}{17} a^{2} - \frac{5}{17}$, $\frac{1}{425} a^{17} - \frac{23}{425} a^{15} - \frac{7}{85} a^{13} + \frac{149}{425} a^{11} - \frac{11}{425} a^{9} + \frac{28}{85} a^{7} - \frac{26}{85} a^{5} + \frac{5}{17} a^{3} - \frac{5}{17} a$, $\frac{1}{3681661042402057726755475} a^{18} - \frac{464934246448079949281}{736332208480411545351095} a^{16} + \frac{182094747154608869562521}{3681661042402057726755475} a^{14} + \frac{94871627397059651533264}{3681661042402057726755475} a^{12} + \frac{1795475868740448678603106}{3681661042402057726755475} a^{10} - \frac{1676988258247312729368853}{3681661042402057726755475} a^{8} - \frac{120446633008432832389847}{736332208480411545351095} a^{6} + \frac{194072132394859968008221}{736332208480411545351095} a^{4} + \frac{613184922626940660417}{3591864431611763635859} a^{2} + \frac{35088377696812286951533}{147266441696082309070219}$, $\frac{1}{18408305212010288633777375} a^{19} + \frac{6338060632235030198902}{18408305212010288633777375} a^{17} - \frac{297962500537829821976346}{3681661042402057726755475} a^{15} + \frac{1264340429101242694149709}{18408305212010288633777375} a^{13} + \frac{8240548375910168557911514}{18408305212010288633777375} a^{11} + \frac{676409430121267671738087}{3681661042402057726755475} a^{9} + \frac{1742040717853784605851158}{3681661042402057726755475} a^{7} + \frac{111581374140565605142223}{736332208480411545351095} a^{5} - \frac{5514113225416656130166}{17959322158058818179295} a^{3} - \frac{60551633003545896183088}{147266441696082309070219} a$

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:  \( 34978022092.6 \) (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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
6029Data not computed