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Properties

Label 20.12.1511207993...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{38}\cdot 5^{20}\cdot 7^{8}$
Root discriminant $40.64$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 20, -100, -520, 2330, 784, -10240, 16560, -18265, 6000, 5036, 1040, -2460, -1280, 330, 296, 95, -20, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 - 20*x^17 + 95*x^16 + 296*x^15 + 330*x^14 - 1280*x^13 - 2460*x^12 + 1040*x^11 + 5036*x^10 + 6000*x^9 - 18265*x^8 + 16560*x^7 - 10240*x^6 + 784*x^5 + 2330*x^4 - 520*x^3 - 100*x^2 + 20*x + 1)
 
gp: K = bnfinit(x^20 - 20*x^18 - 20*x^17 + 95*x^16 + 296*x^15 + 330*x^14 - 1280*x^13 - 2460*x^12 + 1040*x^11 + 5036*x^10 + 6000*x^9 - 18265*x^8 + 16560*x^7 - 10240*x^6 + 784*x^5 + 2330*x^4 - 520*x^3 - 100*x^2 + 20*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{18} - 20 x^{17} + 95 x^{16} + 296 x^{15} + 330 x^{14} - 1280 x^{13} - 2460 x^{12} + 1040 x^{11} + 5036 x^{10} + 6000 x^{9} - 18265 x^{8} + 16560 x^{7} - 10240 x^{6} + 784 x^{5} + 2330 x^{4} - 520 x^{3} - 100 x^{2} + 20 x + 1 \)

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:  \(151120799334400000000000000000000=2^{38}\cdot 5^{20}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.64$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{10} - \frac{2}{5} a^{5} - \frac{2}{5}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{11} - \frac{2}{5} a^{6} - \frac{2}{5} a$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{12} - \frac{2}{5} a^{7} - \frac{2}{5} a^{2}$, $\frac{1}{20} a^{18} - \frac{1}{10} a^{17} + \frac{1}{20} a^{16} + \frac{1}{5} a^{13} + \frac{1}{10} a^{12} + \frac{1}{5} a^{11} - \frac{1}{2} a^{10} - \frac{1}{10} a^{8} + \frac{1}{5} a^{7} - \frac{7}{20} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{2}{5} a^{3} - \frac{1}{20} a^{2} - \frac{1}{10} a - \frac{1}{4}$, $\frac{1}{354842792139898973072357541364420} a^{19} + \frac{59423631315556742414117374793}{35484279213989897307235754136442} a^{18} + \frac{34215136772212802901475619509509}{354842792139898973072357541364420} a^{17} + \frac{7804727797369110763383596274357}{88710698034974743268089385341105} a^{16} - \frac{6333326135715068817681339184082}{88710698034974743268089385341105} a^{15} + \frac{18388962157093811559284297080541}{88710698034974743268089385341105} a^{14} + \frac{1506004005746439784908354441065}{35484279213989897307235754136442} a^{13} + \frac{7791652337273983521994546420214}{88710698034974743268089385341105} a^{12} - \frac{20136386556812464188815131887879}{177421396069949486536178770682210} a^{11} + \frac{23534375573481603017744577995287}{88710698034974743268089385341105} a^{10} - \frac{70919655448113607200340291106651}{177421396069949486536178770682210} a^{9} + \frac{4581095793367492110562993736519}{17742139606994948653617877068221} a^{8} + \frac{7894252741904584843797903473557}{354842792139898973072357541364420} a^{7} - \frac{658468694233127252793086956243}{177421396069949486536178770682210} a^{6} - \frac{168510725707680710915482186452299}{354842792139898973072357541364420} a^{5} + \frac{21948678988066695861375216936422}{88710698034974743268089385341105} a^{4} - \frac{19413726216303254761352979665045}{70968558427979794614471508272884} a^{3} + \frac{68710234094290653430329587507461}{177421396069949486536178770682210} a^{2} - \frac{81229427048864235991270816583161}{354842792139898973072357541364420} a - \frac{13168695352092374601600609975081}{88710698034974743268089385341105}$

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:  \( 1439193689.55 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.15680000000000.1

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 R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$5.10.10.9$x^{10} + 10 x^{8} + 10 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 17$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
5.10.10.9$x^{10} + 10 x^{8} + 10 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 17$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{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}$
7.8.6.3$x^{8} - 7 x^{4} + 147$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$