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Properties

Label 20.0.11412254937...3744.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{32}\cdot 3^{10}\cdot 7^{4}\cdot 37^{4}$
Root discriminant $15.95$
Ramified primes $2, 3, 7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, 16, 0, -4, 0, -44, 0, 77, 0, -54, 0, 29, 0, -2, 0, -7, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^16 - 2*x^14 + 29*x^12 - 54*x^10 + 77*x^8 - 44*x^6 - 4*x^4 + 16*x^2 + 16)
 
gp: K = bnfinit(x^20 - 7*x^16 - 2*x^14 + 29*x^12 - 54*x^10 + 77*x^8 - 44*x^6 - 4*x^4 + 16*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{16} - 2 x^{14} + 29 x^{12} - 54 x^{10} + 77 x^{8} - 44 x^{6} - 4 x^{4} + 16 x^{2} + 16 \)

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:  \(1141225493760614275743744=2^{32}\cdot 3^{10}\cdot 7^{4}\cdot 37^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.95$
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:  $2, 3, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{14} - \frac{7}{32} a^{12} - \frac{1}{4} a^{10} - \frac{7}{32} a^{8} - \frac{1}{2} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{32} a^{4} - \frac{1}{2} a^{3} - \frac{5}{16} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{64} a^{17} - \frac{7}{32} a^{15} - \frac{1}{4} a^{14} + \frac{9}{64} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{7}{64} a^{9} - \frac{5}{16} a^{7} + \frac{1}{4} a^{6} - \frac{15}{64} a^{5} - \frac{5}{32} a^{3} + \frac{1}{4} a^{2} - \frac{3}{16} a$, $\frac{1}{2462144} a^{18} - \frac{19177}{1231072} a^{16} - \frac{1}{4} a^{15} - \frac{255615}{2462144} a^{14} + \frac{130901}{615536} a^{12} - \frac{1}{4} a^{11} + \frac{15865}{2462144} a^{10} - \frac{42087}{307768} a^{8} + \frac{1}{4} a^{7} + \frac{10553}{33728} a^{6} - \frac{194567}{1231072} a^{4} + \frac{1}{4} a^{3} + \frac{101423}{615536} a^{2} - \frac{5261}{153884}$, $\frac{1}{4924288} a^{19} - \frac{19177}{2462144} a^{17} + \frac{975457}{4924288} a^{15} - \frac{1}{4} a^{14} - \frac{176867}{1231072} a^{13} + \frac{15865}{4924288} a^{11} - \frac{1}{4} a^{10} + \frac{111797}{615536} a^{9} + \frac{10553}{67456} a^{7} - \frac{1}{4} a^{6} - \frac{194567}{2462144} a^{5} - \frac{1}{2} a^{4} - \frac{206345}{1231072} a^{3} - \frac{1}{4} a^{2} - \frac{5261}{307768} a - \frac{1}{2}$

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:  \( -\frac{63}{4964} a^{18} + \frac{75}{4964} a^{16} + \frac{529}{4964} a^{14} - \frac{31}{4964} a^{12} - \frac{1731}{4964} a^{10} + \frac{4399}{4964} a^{8} - \frac{139}{68} a^{6} + \frac{6933}{4964} a^{4} - \frac{981}{1241} a^{2} + \frac{384}{1241} \) (order $6$)
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:  \( 21174.0969976 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.0.16691899392.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.12.20.28$x^{12} - 4 x^{10} + x^{8} + 2 x^{6} - 7 x^{4} + 2 x^{2} + 1$$6$$2$$20$12T158$[2, 8/3, 8/3, 8/3, 8/3]_{3}^{6}$
3Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.3.2.3$x^{3} - 148$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.3$x^{3} - 148$$3$$1$$2$$C_3$$[\ ]_{3}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$