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Properties

Label 20.8.69269022970...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{57}\cdot 5^{22}\cdot 17^{10}$
Root discriminant $174.59$
Ramified primes $2, 5, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![167811200, 0, -231354400, 0, 93816450, 0, -11759900, 0, -346600, 0, 315400, 0, -15105, 0, -2400, 0, -80, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 80*x^16 - 2400*x^14 - 15105*x^12 + 315400*x^10 - 346600*x^8 - 11759900*x^6 + 93816450*x^4 - 231354400*x^2 + 167811200)
 
gp: K = bnfinit(x^20 - 80*x^16 - 2400*x^14 - 15105*x^12 + 315400*x^10 - 346600*x^8 - 11759900*x^6 + 93816450*x^4 - 231354400*x^2 + 167811200, 1)
 

Normalized defining polynomial

\( x^{20} - 80 x^{16} - 2400 x^{14} - 15105 x^{12} + 315400 x^{10} - 346600 x^{8} - 11759900 x^{6} + 93816450 x^{4} - 231354400 x^{2} + 167811200 \)

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:  \(692690229709114777272320000000000000000000000=2^{57}\cdot 5^{22}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $174.59$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10}$, $\frac{1}{5} a^{11}$, $\frac{1}{5} a^{12}$, $\frac{1}{5} a^{13}$, $\frac{1}{5} a^{14}$, $\frac{1}{5} a^{15}$, $\frac{1}{10} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{20} a^{17} - \frac{1}{4} a^{9} - \frac{1}{2} a$, $\frac{1}{25144078135498382255277328966731520} a^{18} - \frac{30613934528363079353302761124163}{628601953387459556381933224168288} a^{16} - \frac{30980361825931377975177473878289}{1571504883468648890954833060420720} a^{14} - \frac{37371758624342109344214821618271}{785752441734324445477416530210360} a^{12} + \frac{1723266556454282800005957751684351}{25144078135498382255277328966731520} a^{10} + \frac{10072216163849311572367062628907}{39287622086716222273870826510518} a^{8} + \frac{66401842514338928251172395465543}{628601953387459556381933224168288} a^{6} + \frac{82867651791711919755860930376597}{1257203906774919112763866448336576} a^{4} + \frac{218492259347247084712228452163037}{2514407813549838225527732896673152} a^{2} - \frac{132033774133015114858483635685645}{314300976693729778190966612084144}$, $\frac{1}{11515987786058259072917016666763036160} a^{19} - \frac{153015188146924981554427108065331}{6286019533874595563819332241682880} a^{17} - \frac{12515374944775631447595506655724589}{143949847325728238411462708334537952} a^{15} - \frac{1013237477144803712079936122992915}{71974923662864119205731354167268976} a^{13} + \frac{61696203961906909262877874229161779}{2303197557211651814583403333352607232} a^{11} + \frac{3865044978101793494193992235972847}{8996865457858014900716419270908622} a^{9} - \frac{128796998601914870130045138559033497}{287899694651456476822925416669075904} a^{7} - \frac{173411271483147125641657708940070891}{575799389302912953645850833338151808} a^{5} - \frac{37497624943900326298203764997934243}{1151598778605825907291701666676303616} a^{3} + \frac{54242035193882236512178740254871267}{143949847325728238411462708334537952} a$

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:  \( 5020245578500000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 10.6.1817416960000000000.1

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 32 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ $16{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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]$
2.8.25.53$x^{8} + 4 x^{6} + 104$$8$$1$$25$$C_2 \wr S_4$$[8/3, 8/3, 3, 23/6, 23/6, 17/4]_{3}^{2}$
2.8.26.108$x^{8} + 8 x^{6} + 336$$8$$1$$26$$C_2 \wr C_2\wr C_2$$[2, 2, 3, 7/2, 4, 17/4]^{2}$
5Data not computed
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$