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Properties

Label 20.8.12047257600...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{32}\cdot 5^{22}\cdot 7^{6}$
Root discriminant $31.92$
Ramified primes $2, 5, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 960, -8620, 4360, 24195, -5224, -21200, 3710, 9105, -4250, -1903, 3200, 50, -1150, -80, 254, 35, -20, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 - 20*x^17 + 35*x^16 + 254*x^15 - 80*x^14 - 1150*x^13 + 50*x^12 + 3200*x^11 - 1903*x^10 - 4250*x^9 + 9105*x^8 + 3710*x^7 - 21200*x^6 - 5224*x^5 + 24195*x^4 + 4360*x^3 - 8620*x^2 + 960*x - 4)
 
gp: K = bnfinit(x^20 - 10*x^18 - 20*x^17 + 35*x^16 + 254*x^15 - 80*x^14 - 1150*x^13 + 50*x^12 + 3200*x^11 - 1903*x^10 - 4250*x^9 + 9105*x^8 + 3710*x^7 - 21200*x^6 - 5224*x^5 + 24195*x^4 + 4360*x^3 - 8620*x^2 + 960*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} - 20 x^{17} + 35 x^{16} + 254 x^{15} - 80 x^{14} - 1150 x^{13} + 50 x^{12} + 3200 x^{11} - 1903 x^{10} - 4250 x^{9} + 9105 x^{8} + 3710 x^{7} - 21200 x^{6} - 5224 x^{5} + 24195 x^{4} + 4360 x^{3} - 8620 x^{2} + 960 x - 4 \)

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:  \(1204725760000000000000000000000=2^{32}\cdot 5^{22}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.92$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{8} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} + \frac{3}{8} a^{14} + \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{3}{8} a^{8} + \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{1050262311267406101462482664981959174776} a^{19} - \frac{3035382329589778047159167782930257711}{262565577816851525365620666245489793694} a^{18} + \frac{65631176920231440241166793677875409225}{131282788908425762682810333122744896847} a^{17} - \frac{16081127559325283804600160533868300087}{262565577816851525365620666245489793694} a^{16} + \frac{477965171635883155890150933501128659291}{1050262311267406101462482664981959174776} a^{15} + \frac{233852474491309618591679747204747410657}{525131155633703050731241332490979587388} a^{14} + \frac{94061551938494298933328322912407992823}{525131155633703050731241332490979587388} a^{13} + \frac{219175402112332595780516706456033871671}{525131155633703050731241332490979587388} a^{12} - \frac{149536074228449022316479738318135765597}{525131155633703050731241332490979587388} a^{11} + \frac{126334737463456238246447114292645760649}{262565577816851525365620666245489793694} a^{10} + \frac{27028323208936096515469147772712223309}{1050262311267406101462482664981959174776} a^{9} + \frac{22120136319823241979927651791970761653}{525131155633703050731241332490979587388} a^{8} - \frac{38663785100480672178357039712999909429}{1050262311267406101462482664981959174776} a^{7} - \frac{218018515012053291563861120196016217217}{525131155633703050731241332490979587388} a^{6} + \frac{143896466023559904733625774055379570987}{525131155633703050731241332490979587388} a^{5} - \frac{122398778620307169508743943460243315283}{262565577816851525365620666245489793694} a^{4} + \frac{127696838298051752099538349356098378815}{1050262311267406101462482664981959174776} a^{3} - \frac{70148193210788089131456975995515679967}{262565577816851525365620666245489793694} a^{2} - \frac{78675184403435245181122386231721677591}{525131155633703050731241332490979587388} a - \frac{21528433859346750606294000954991953323}{131282788908425762682810333122744896847}$

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:  \( 36223048.1655 \) (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 ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$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}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$