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Properties

Label 20.8.35762115898...0000.6
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 3^{8}\cdot 5^{11}\cdot 239^{8}$
Root discriminant $67.25$
Ramified primes $2, 3, 5, 239$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![605, 0, 2825, 0, -1054, 0, -10487, 0, 2330, 0, 3958, 0, -226, 0, -413, 0, -10, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 12*x^18 - 10*x^16 - 413*x^14 - 226*x^12 + 3958*x^10 + 2330*x^8 - 10487*x^6 - 1054*x^4 + 2825*x^2 + 605)
 
gp: K = bnfinit(x^20 + 12*x^18 - 10*x^16 - 413*x^14 - 226*x^12 + 3958*x^10 + 2330*x^8 - 10487*x^6 - 1054*x^4 + 2825*x^2 + 605, 1)
 

Normalized defining polynomial

\( x^{20} + 12 x^{18} - 10 x^{16} - 413 x^{14} - 226 x^{12} + 3958 x^{10} + 2330 x^{8} - 10487 x^{6} - 1054 x^{4} + 2825 x^{2} + 605 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3576211589862019840319539200000000000=2^{20}\cdot 3^{8}\cdot 5^{11}\cdot 239^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.25$
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:  $2, 3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{4}{9} a^{4} + \frac{4}{9} a^{2} - \frac{1}{9}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{4}{9} a^{5} + \frac{4}{9} a^{3} - \frac{1}{9} a$, $\frac{1}{135} a^{12} - \frac{4}{135} a^{10} - \frac{2}{15} a^{8} - \frac{4}{15} a^{6} - \frac{4}{135} a^{4} - \frac{11}{27}$, $\frac{1}{135} a^{13} - \frac{4}{135} a^{11} - \frac{2}{15} a^{9} - \frac{4}{15} a^{7} - \frac{4}{135} a^{5} - \frac{11}{27} a$, $\frac{1}{135} a^{14} - \frac{4}{135} a^{10} + \frac{4}{45} a^{8} - \frac{43}{135} a^{6} + \frac{14}{135} a^{4} - \frac{5}{27} a^{2} + \frac{13}{27}$, $\frac{1}{135} a^{15} - \frac{4}{135} a^{11} + \frac{4}{45} a^{9} - \frac{43}{135} a^{7} + \frac{14}{135} a^{5} - \frac{5}{27} a^{3} + \frac{13}{27} a$, $\frac{1}{1215} a^{16} + \frac{1}{1215} a^{14} - \frac{1}{1215} a^{12} - \frac{4}{1215} a^{10} - \frac{8}{243} a^{8} + \frac{538}{1215} a^{6} + \frac{472}{1215} a^{4} + \frac{26}{243} a^{2} + \frac{52}{243}$, $\frac{1}{1215} a^{17} + \frac{1}{1215} a^{15} - \frac{1}{1215} a^{13} - \frac{4}{1215} a^{11} - \frac{8}{243} a^{9} + \frac{538}{1215} a^{7} + \frac{472}{1215} a^{5} + \frac{26}{243} a^{3} + \frac{52}{243} a$, $\frac{1}{617634315} a^{18} + \frac{46613}{617634315} a^{16} - \frac{48559}{68626035} a^{14} - \frac{1941584}{617634315} a^{12} - \frac{22000283}{617634315} a^{10} - \frac{29260484}{205878105} a^{8} - \frac{58506613}{617634315} a^{6} - \frac{138331792}{617634315} a^{4} + \frac{10842911}{41175621} a^{2} - \frac{18658496}{123526863}$, $\frac{1}{6793977465} a^{19} - \frac{295682}{1358795493} a^{17} - \frac{726347}{754886385} a^{15} - \frac{998326}{1358795493} a^{13} + \frac{89326396}{6793977465} a^{11} + \frac{23098639}{2264659155} a^{9} + \frac{31469744}{6793977465} a^{7} + \frac{2458782377}{6793977465} a^{5} - \frac{141151048}{452931831} a^{3} - \frac{614942489}{1358795493} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 47292286187.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
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}$
5.4.2.1$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}$
239Data not computed