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Properties

Label 20.2.12868967127...3856.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{30}\cdot 79^{9}$
Root discriminant $20.21$
Ramified primes $2, 79$
Class number $1$
Class group Trivial
Galois group $C_5:D_4$ (as 20T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 58, -148, 309, -320, 424, -88, 88, 312, -154, 172, -99, -68, 22, -20, 15, 8, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^17 + 15*x^16 - 20*x^15 + 22*x^14 - 68*x^13 - 99*x^12 + 172*x^11 - 154*x^10 + 312*x^9 + 88*x^8 - 88*x^7 + 424*x^6 - 320*x^5 + 309*x^4 - 148*x^3 + 58*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^17 + 15*x^16 - 20*x^15 + 22*x^14 - 68*x^13 - 99*x^12 + 172*x^11 - 154*x^10 + 312*x^9 + 88*x^8 - 88*x^7 + 424*x^6 - 320*x^5 + 309*x^4 - 148*x^3 + 58*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{17} + 15 x^{16} - 20 x^{15} + 22 x^{14} - 68 x^{13} - 99 x^{12} + 172 x^{11} - 154 x^{10} + 312 x^{9} + 88 x^{8} - 88 x^{7} + 424 x^{6} - 320 x^{5} + 309 x^{4} - 148 x^{3} + 58 x^{2} - 12 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-128689671279687666138873856=-\,2^{30}\cdot 79^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.21$
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:  $2, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{42} a^{18} + \frac{2}{21} a^{17} - \frac{1}{7} a^{16} - \frac{1}{14} a^{15} + \frac{3}{14} a^{14} - \frac{1}{21} a^{13} + \frac{4}{21} a^{11} + \frac{1}{6} a^{10} - \frac{13}{42} a^{9} + \frac{1}{42} a^{8} - \frac{5}{42} a^{7} - \frac{11}{42} a^{6} + \frac{1}{3} a^{5} + \frac{3}{14} a^{4} - \frac{1}{14} a^{3} - \frac{5}{14} a^{2} + \frac{1}{3} a + \frac{5}{42}$, $\frac{1}{5957845134985911174} a^{19} + \frac{2382724735569335}{283706911189805294} a^{18} - \frac{733946418440843087}{2978922567492955587} a^{17} - \frac{36296045889224675}{283706911189805294} a^{16} - \frac{21790848232856661}{141853455594902647} a^{15} - \frac{152077108979519260}{2978922567492955587} a^{14} - \frac{379953353561745260}{2978922567492955587} a^{13} + \frac{85274690000145563}{5957845134985911174} a^{12} + \frac{1332567555751376171}{5957845134985911174} a^{11} - \frac{114579627832625749}{2978922567492955587} a^{10} + \frac{692314868970578947}{2978922567492955587} a^{9} - \frac{358635017964217907}{1985948378328637058} a^{8} + \frac{267498095538280598}{992974189164318529} a^{7} + \frac{2183335717971227941}{5957845134985911174} a^{6} + \frac{1797574305753012229}{5957845134985911174} a^{5} - \frac{37889309990261869}{1985948378328637058} a^{4} - \frac{425439873810904488}{992974189164318529} a^{3} + \frac{1089753474239577289}{2978922567492955587} a^{2} - \frac{266275995327563665}{1985948378328637058} a + \frac{532318056666128093}{2978922567492955587}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 107328.34068 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_4$ (as 20T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.5056.1, 5.1.6241.1, 10.2.1276316254208.1

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

Sibling fields

Galois closure: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$