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Properties

Label 20.16.1853091914...0000.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{20}\cdot 5^{14}\cdot 6029^{7}$
Root discriminant $129.84$
Ramified primes $2, 5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![219147162389, 0, -245027537181, 0, 81314178075, 0, -3026975115, 0, -2468969743, 0, 325190591, 0, -13030762, 0, 26505, 0, 9050, 0, -181, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 181*x^18 + 9050*x^16 + 26505*x^14 - 13030762*x^12 + 325190591*x^10 - 2468969743*x^8 - 3026975115*x^6 + 81314178075*x^4 - 245027537181*x^2 + 219147162389)
 
gp: K = bnfinit(x^20 - 181*x^18 + 9050*x^16 + 26505*x^14 - 13030762*x^12 + 325190591*x^10 - 2468969743*x^8 - 3026975115*x^6 + 81314178075*x^4 - 245027537181*x^2 + 219147162389, 1)
 

Normalized defining polynomial

\( x^{20} - 181 x^{18} + 9050 x^{16} + 26505 x^{14} - 13030762 x^{12} + 325190591 x^{10} - 2468969743 x^{8} - 3026975115 x^{6} + 81314178075 x^{4} - 245027537181 x^{2} + 219147162389 \)

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:  \(1853091914135141046229493177600000000000000=2^{20}\cdot 5^{14}\cdot 6029^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.84$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{6029} a^{16} - \frac{181}{6029} a^{14} - \frac{3008}{6029} a^{12} + \frac{2389}{6029} a^{10} - \frac{2093}{6029} a^{8} - \frac{1611}{6029} a^{6} + \frac{2221}{6029} a^{4} - \frac{1114}{6029} a^{2}$, $\frac{1}{6029} a^{17} - \frac{181}{6029} a^{15} - \frac{3008}{6029} a^{13} + \frac{2389}{6029} a^{11} - \frac{2093}{6029} a^{9} - \frac{1611}{6029} a^{7} + \frac{2221}{6029} a^{5} - \frac{1114}{6029} a^{3}$, $\frac{1}{307300843648045682465457978430014128458220483997649271} a^{18} + \frac{1447455268519128568356363974049927078780967416}{223817074761868668947893647800447289481588116531427} a^{16} - \frac{87155617822440268913791321327653467070390026578204081}{307300843648045682465457978430014128458220483997649271} a^{14} - \frac{150829179045424701589463122460733004970153048246642116}{307300843648045682465457978430014128458220483997649271} a^{12} - \frac{84313432924575588119427603127619773879993491351135001}{307300843648045682465457978430014128458220483997649271} a^{10} - \frac{143688919744943576202612221913549572742220646044421636}{307300843648045682465457978430014128458220483997649271} a^{8} + \frac{64733972473787327563900414793410303700581919107908166}{307300843648045682465457978430014128458220483997649271} a^{6} - \frac{127293355882629388546926282446635963133465407763819036}{307300843648045682465457978430014128458220483997649271} a^{4} - \frac{14957058628065693913700239277362348796325779966622}{50970450099194838690571898893682887453677307015699} a^{2} + \frac{3956259410534582416416312771207411799374800039}{8454212987094848016349626620282449403495987231}$, $\frac{1}{307300843648045682465457978430014128458220483997649271} a^{19} + \frac{1447455268519128568356363974049927078780967416}{223817074761868668947893647800447289481588116531427} a^{17} - \frac{87155617822440268913791321327653467070390026578204081}{307300843648045682465457978430014128458220483997649271} a^{15} - \frac{150829179045424701589463122460733004970153048246642116}{307300843648045682465457978430014128458220483997649271} a^{13} - \frac{84313432924575588119427603127619773879993491351135001}{307300843648045682465457978430014128458220483997649271} a^{11} - \frac{143688919744943576202612221913549572742220646044421636}{307300843648045682465457978430014128458220483997649271} a^{9} + \frac{64733972473787327563900414793410303700581919107908166}{307300843648045682465457978430014128458220483997649271} a^{7} - \frac{127293355882629388546926282446635963133465407763819036}{307300843648045682465457978430014128458220483997649271} a^{5} - \frac{14957058628065693913700239277362348796325779966622}{50970450099194838690571898893682887453677307015699} a^{3} + \frac{3956259410534582416416312771207411799374800039}{8454212987094848016349626620282449403495987231} a$

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:  $17$
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:  \( 183209198928000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 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 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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
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}$
6029Data not computed