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Properties

Label 20.0.41384987168...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{46}\cdot 5^{12}\cdot 7^{4}\cdot 79\cdot 127$
Root discriminant $30.26$
Ramified primes $2, 5, 7, 79, 127$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T925

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![56, -200, 750, -1360, 3425, -1512, 1100, -480, -290, -40, -226, 0, 115, 0, 120, 0, 50, 0, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 10*x^18 + 50*x^16 + 120*x^14 + 115*x^12 - 226*x^10 - 40*x^9 - 290*x^8 - 480*x^7 + 1100*x^6 - 1512*x^5 + 3425*x^4 - 1360*x^3 + 750*x^2 - 200*x + 56)
 
gp: K = bnfinit(x^20 + 10*x^18 + 50*x^16 + 120*x^14 + 115*x^12 - 226*x^10 - 40*x^9 - 290*x^8 - 480*x^7 + 1100*x^6 - 1512*x^5 + 3425*x^4 - 1360*x^3 + 750*x^2 - 200*x + 56, 1)
 

Normalized defining polynomial

\( x^{20} + 10 x^{18} + 50 x^{16} + 120 x^{14} + 115 x^{12} - 226 x^{10} - 40 x^{9} - 290 x^{8} - 480 x^{7} + 1100 x^{6} - 1512 x^{5} + 3425 x^{4} - 1360 x^{3} + 750 x^{2} - 200 x + 56 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(413849871682895872000000000000=2^{46}\cdot 5^{12}\cdot 7^{4}\cdot 79\cdot 127\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.26$
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:  $2, 5, 7, 79, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{16} + \frac{1}{20} a^{14} + \frac{1}{10} a^{13} - \frac{1}{20} a^{11} + \frac{3}{20} a^{10} + \frac{1}{10} a^{9} + \frac{3}{10} a^{8} - \frac{1}{20} a^{7} - \frac{9}{20} a^{6} - \frac{3}{10} a^{5} + \frac{3}{20} a^{4} - \frac{3}{20} a^{3} + \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{20} a^{17} + \frac{1}{20} a^{15} + \frac{1}{10} a^{14} - \frac{1}{20} a^{12} - \frac{1}{10} a^{11} + \frac{1}{10} a^{10} + \frac{3}{10} a^{9} - \frac{1}{20} a^{8} + \frac{3}{10} a^{7} - \frac{3}{10} a^{6} + \frac{3}{20} a^{5} - \frac{3}{20} a^{4} + \frac{1}{4} a^{3} + \frac{1}{10} a^{2} + \frac{3}{10} a$, $\frac{1}{20} a^{18} + \frac{1}{10} a^{15} - \frac{1}{20} a^{14} + \frac{1}{10} a^{13} - \frac{1}{10} a^{12} - \frac{1}{10} a^{11} + \frac{3}{20} a^{10} + \frac{1}{10} a^{9} - \frac{1}{2} a^{7} - \frac{2}{5} a^{6} - \frac{1}{10} a^{5} + \frac{1}{10} a^{4} + \frac{3}{10} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{1375767025240220746012860} a^{19} + \frac{33663598811722522751363}{1375767025240220746012860} a^{18} + \frac{3408662274741408356051}{137576702524022074601286} a^{17} + \frac{7880340908600607425083}{343941756310055186503215} a^{16} + \frac{9955231534710197517533}{275153405048044149202572} a^{15} - \frac{19604341713887550235114}{343941756310055186503215} a^{14} - \frac{12726879016923271219139}{343941756310055186503215} a^{13} - \frac{13098336357600079251112}{343941756310055186503215} a^{12} + \frac{146912270503880450063717}{1375767025240220746012860} a^{11} + \frac{10897196774137258702249}{343941756310055186503215} a^{10} + \frac{56557456336388270584043}{687883512620110373006430} a^{9} + \frac{37599461244786199303459}{137576702524022074601286} a^{8} - \frac{36503441831195896763463}{229294504206703457668810} a^{7} - \frac{90923880947475218263977}{458589008413406915337620} a^{6} + \frac{152511331508139253083943}{687883512620110373006430} a^{5} - \frac{283485237670686948817867}{687883512620110373006430} a^{4} - \frac{164387170569489066238657}{687883512620110373006430} a^{3} - \frac{144778950733844761998967}{687883512620110373006430} a^{2} - \frac{119712830085060361179568}{343941756310055186503215} a + \frac{148016548711112209391168}{343941756310055186503215}$

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:  $9$
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:  \( 44919957.3839 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T925:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 409600
The 190 conjugacy class representatives for t20n925 are not computed
Character table for t20n925 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.2.6422528000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.20.4$x^{8} + 72 x^{4} + 656$$4$$2$$20$$Q_8:C_2$$[2, 3, 7/2]^{2}$
2.8.20.4$x^{8} + 72 x^{4} + 656$$4$$2$$20$$Q_8:C_2$$[2, 3, 7/2]^{2}$
5Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
79Data not computed
$127$127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
127.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
127.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
127.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$