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Properties

Label 20.14.8151193332...1875.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,3^{8}\cdot 5^{11}\cdot 239^{9}$
Root discriminant $44.21$
Ramified primes $3, 5, 239$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 66, 187, -1500, -1956, 3547, 6396, -47, -8745, -5813, 5736, 4195, -2757, -298, 1345, -187, -166, 83, 0, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 83*x^17 - 166*x^16 - 187*x^15 + 1345*x^14 - 298*x^13 - 2757*x^12 + 4195*x^11 + 5736*x^10 - 5813*x^9 - 8745*x^8 - 47*x^7 + 6396*x^6 + 3547*x^5 - 1956*x^4 - 1500*x^3 + 187*x^2 + 66*x - 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 83*x^17 - 166*x^16 - 187*x^15 + 1345*x^14 - 298*x^13 - 2757*x^12 + 4195*x^11 + 5736*x^10 - 5813*x^9 - 8745*x^8 - 47*x^7 + 6396*x^6 + 3547*x^5 - 1956*x^4 - 1500*x^3 + 187*x^2 + 66*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 83 x^{17} - 166 x^{16} - 187 x^{15} + 1345 x^{14} - 298 x^{13} - 2757 x^{12} + 4195 x^{11} + 5736 x^{10} - 5813 x^{9} - 8745 x^{8} - 47 x^{7} + 6396 x^{6} + 3547 x^{5} - 1956 x^{4} - 1500 x^{3} + 187 x^{2} + 66 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-815119333245299093090410107421875=-\,3^{8}\cdot 5^{11}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.21$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{16} - \frac{4}{9} a^{13} - \frac{1}{3} a^{12} - \frac{2}{9} a^{11} - \frac{2}{9} a^{10} - \frac{1}{3} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{283521446904460524516208417977} a^{19} + \frac{3193192549104560002484477135}{94507148968153508172069472659} a^{18} + \frac{45936280079558388071852627326}{283521446904460524516208417977} a^{17} - \frac{2543705187586406353446224223}{31502382989384502724023157553} a^{16} + \frac{3769307656962887755959513184}{31502382989384502724023157553} a^{15} + \frac{30486751445562248503237489001}{283521446904460524516208417977} a^{14} + \frac{38847726470974433048472613186}{94507148968153508172069472659} a^{13} + \frac{132029378033006715371376341149}{283521446904460524516208417977} a^{12} - \frac{112362274566473381905428318518}{283521446904460524516208417977} a^{11} + \frac{8636528143367393985765082526}{94507148968153508172069472659} a^{10} + \frac{96087344834853659204431642318}{283521446904460524516208417977} a^{9} - \frac{14222844489994470041278067527}{283521446904460524516208417977} a^{8} + \frac{14779603442243565909979841271}{31502382989384502724023157553} a^{7} + \frac{23158105158902509094464716701}{94507148968153508172069472659} a^{6} - \frac{41772023703244836909445066789}{94507148968153508172069472659} a^{5} + \frac{76605616344227103569092718554}{283521446904460524516208417977} a^{4} - \frac{18185975641320328965342008543}{283521446904460524516208417977} a^{3} - \frac{47177678568460528718397743740}{283521446904460524516208417977} a^{2} - \frac{123424877236137381875131687649}{283521446904460524516208417977} a + \frac{15891612467063439483183538733}{94507148968153508172069472659}$

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:  $16$
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:  \( 2453065917.48 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 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 $20$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.4.2.1$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$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed