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Properties

Label 20.20.2618179372...5625.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{10}\cdot 401^{9}$
Root discriminant $33.18$
Ramified primes $5, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
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, -100, 846, -514, -4819, 5587, 9640, -14287, -8756, 16864, 3697, -10913, -396, 4107, -253, -894, 108, 104, -17, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 17*x^18 + 104*x^17 + 108*x^16 - 894*x^15 - 253*x^14 + 4107*x^13 - 396*x^12 - 10913*x^11 + 3697*x^10 + 16864*x^9 - 8756*x^8 - 14287*x^7 + 9640*x^6 + 5587*x^5 - 4819*x^4 - 514*x^3 + 846*x^2 - 100*x - 1)
 
gp: K = bnfinit(x^20 - 5*x^19 - 17*x^18 + 104*x^17 + 108*x^16 - 894*x^15 - 253*x^14 + 4107*x^13 - 396*x^12 - 10913*x^11 + 3697*x^10 + 16864*x^9 - 8756*x^8 - 14287*x^7 + 9640*x^6 + 5587*x^5 - 4819*x^4 - 514*x^3 + 846*x^2 - 100*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 17 x^{18} + 104 x^{17} + 108 x^{16} - 894 x^{15} - 253 x^{14} + 4107 x^{13} - 396 x^{12} - 10913 x^{11} + 3697 x^{10} + 16864 x^{9} - 8756 x^{8} - 14287 x^{7} + 9640 x^{6} + 5587 x^{5} - 4819 x^{4} - 514 x^{3} + 846 x^{2} - 100 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2618179372631552556285166015625=5^{10}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.18$
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:  $5, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{412736007} a^{19} - \frac{22273359}{137578669} a^{18} - \frac{48179528}{412736007} a^{17} - \frac{11804288}{137578669} a^{16} + \frac{26436718}{412736007} a^{15} - \frac{31599371}{412736007} a^{14} - \frac{5241430}{412736007} a^{13} - \frac{49478212}{412736007} a^{12} - \frac{172814542}{412736007} a^{11} - \frac{124971746}{412736007} a^{10} + \frac{201097600}{412736007} a^{9} + \frac{30813046}{137578669} a^{8} - \frac{4247681}{137578669} a^{7} + \frac{21284860}{412736007} a^{6} - \frac{66144164}{412736007} a^{5} - \frac{46820557}{137578669} a^{4} - \frac{66375890}{412736007} a^{3} + \frac{153812564}{412736007} a^{2} - \frac{148294243}{412736007} a + \frac{38206945}{412736007}$

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:  $19$
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:  \( 481290724.965 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 4.4.10025.1, 5.5.160801.1, 10.10.80803005003125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
401Data not computed