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Properties

Label 20.0.86204604797...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{35}\cdot 7^{10}$
Root discriminant $88.47$
Ramified primes $2, 5, 7$
Class number $587584$ (GRH)
Class group $[2, 2, 2, 2, 2, 18362]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1412376245, 0, 5044200875, 0, 5044200875, 0, 2202977525, 0, 505890700, 0, 67312035, 0, 5462275, 0, 274400, 0, 8330, 0, 140, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 140*x^18 + 8330*x^16 + 274400*x^14 + 5462275*x^12 + 67312035*x^10 + 505890700*x^8 + 2202977525*x^6 + 5044200875*x^4 + 5044200875*x^2 + 1412376245)
 
gp: K = bnfinit(x^20 + 140*x^18 + 8330*x^16 + 274400*x^14 + 5462275*x^12 + 67312035*x^10 + 505890700*x^8 + 2202977525*x^6 + 5044200875*x^4 + 5044200875*x^2 + 1412376245, 1)
 

Normalized defining polynomial

\( x^{20} + 140 x^{18} + 8330 x^{16} + 274400 x^{14} + 5462275 x^{12} + 67312035 x^{10} + 505890700 x^{8} + 2202977525 x^{6} + 5044200875 x^{4} + 5044200875 x^{2} + 1412376245 \)

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:  \(862046047973632812500000000000000000000=2^{20}\cdot 5^{35}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.47$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(700=2^{2}\cdot 5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{700}(1,·)$, $\chi_{700}(643,·)$, $\chi_{700}(449,·)$, $\chi_{700}(587,·)$, $\chi_{700}(141,·)$, $\chi_{700}(589,·)$, $\chi_{700}(83,·)$, $\chi_{700}(281,·)$, $\chi_{700}(27,·)$, $\chi_{700}(29,·)$, $\chi_{700}(223,·)$, $\chi_{700}(421,·)$, $\chi_{700}(167,·)$, $\chi_{700}(169,·)$, $\chi_{700}(363,·)$, $\chi_{700}(561,·)$, $\chi_{700}(307,·)$, $\chi_{700}(309,·)$, $\chi_{700}(503,·)$, $\chi_{700}(447,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{49} a^{4}$, $\frac{1}{49} a^{5}$, $\frac{1}{343} a^{6}$, $\frac{1}{343} a^{7}$, $\frac{1}{2401} a^{8}$, $\frac{1}{2401} a^{9}$, $\frac{1}{16807} a^{10}$, $\frac{1}{16807} a^{11}$, $\frac{1}{117649} a^{12}$, $\frac{1}{117649} a^{13}$, $\frac{1}{823543} a^{14}$, $\frac{1}{823543} a^{15}$, $\frac{1}{5764801} a^{16}$, $\frac{1}{5764801} a^{17}$, $\frac{1}{40353607} a^{18}$, $\frac{1}{40353607} a^{19}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{18362}$, which has order $587584$ (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:  \( -1 \) (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:  \( 161406.837641 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.98000.1, 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$