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Properties

Label 20.0.10675123826...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{34}\cdot 7^{10}$
Root discriminant $282.76$
Ramified primes $2, 3, 5, 7$
Class number $347426816$ (GRH)
Class group $[2, 2, 4, 4, 4, 44, 30844]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7061881225, 0, 40353607000, 0, 64854011250, 0, 42000693000, 0, 12356086225, 0, 1775155340, 0, 131334700, 0, 4973500, 0, 88200, 0, 560, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 560*x^18 + 88200*x^16 + 4973500*x^14 + 131334700*x^12 + 1775155340*x^10 + 12356086225*x^8 + 42000693000*x^6 + 64854011250*x^4 + 40353607000*x^2 + 7061881225)
 
gp: K = bnfinit(x^20 + 560*x^18 + 88200*x^16 + 4973500*x^14 + 131334700*x^12 + 1775155340*x^10 + 12356086225*x^8 + 42000693000*x^6 + 64854011250*x^4 + 40353607000*x^2 + 7061881225, 1)
 

Normalized defining polynomial

\( x^{20} + 560 x^{18} + 88200 x^{16} + 4973500 x^{14} + 131334700 x^{12} + 1775155340 x^{10} + 12356086225 x^{8} + 42000693000 x^{6} + 64854011250 x^{4} + 40353607000 x^{2} + 7061881225 \)

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:  \(10675123826048640000000000000000000000000000000000=2^{40}\cdot 3^{10}\cdot 5^{34}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $282.76$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4200=2^{3}\cdot 3\cdot 5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{4200}(1,·)$, $\chi_{4200}(3779,·)$, $\chi_{4200}(71,·)$, $\chi_{4200}(841,·)$, $\chi_{4200}(911,·)$, $\chi_{4200}(1681,·)$, $\chi_{4200}(1751,·)$, $\chi_{4200}(2521,·)$, $\chi_{4200}(349,·)$, $\chi_{4200}(2591,·)$, $\chi_{4200}(3361,·)$, $\chi_{4200}(419,·)$, $\chi_{4200}(1189,·)$, $\chi_{4200}(3431,·)$, $\chi_{4200}(1259,·)$, $\chi_{4200}(2029,·)$, $\chi_{4200}(2099,·)$, $\chi_{4200}(2869,·)$, $\chi_{4200}(2939,·)$, $\chi_{4200}(3709,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{49} a^{4}$, $\frac{1}{49} a^{5}$, $\frac{1}{343} a^{6}$, $\frac{1}{343} a^{7}$, $\frac{1}{2401} a^{8}$, $\frac{1}{2401} a^{9}$, $\frac{1}{84035} a^{10}$, $\frac{1}{84035} a^{11}$, $\frac{1}{4117715} a^{12} - \frac{2}{2401} a^{6} + \frac{3}{7}$, $\frac{1}{4117715} a^{13} - \frac{2}{2401} a^{7} + \frac{3}{7} a$, $\frac{1}{28824005} a^{14} - \frac{2}{16807} a^{8} + \frac{3}{49} a^{2}$, $\frac{1}{28824005} a^{15} - \frac{2}{16807} a^{9} + \frac{3}{49} a^{3}$, $\frac{1}{1412376245} a^{16} - \frac{2}{201768035} a^{14} + \frac{2}{28824005} a^{12} - \frac{17}{4117715} a^{10} - \frac{3}{117649} a^{8} + \frac{3}{16807} a^{6} - \frac{11}{2401} a^{4} + \frac{22}{343} a^{2} - \frac{22}{49}$, $\frac{1}{1412376245} a^{17} - \frac{2}{201768035} a^{15} + \frac{2}{28824005} a^{13} - \frac{17}{4117715} a^{11} - \frac{3}{117649} a^{9} + \frac{3}{16807} a^{7} - \frac{11}{2401} a^{5} + \frac{22}{343} a^{3} - \frac{22}{49} a$, $\frac{1}{13307626001886677965} a^{18} - \frac{148902326}{1901089428840953995} a^{16} + \frac{85596927}{271584204120136285} a^{14} + \frac{2505305169}{38797743445733755} a^{12} - \frac{25812550767}{5542534777961965} a^{10} - \frac{14201857592}{158358136513199} a^{8} - \frac{24515505233}{22622590930457} a^{6} - \frac{16043783557}{3231798704351} a^{4} - \frac{22099821665}{461685529193} a^{2} - \frac{23152528759}{65955075599}$, $\frac{1}{13307626001886677965} a^{19} - \frac{148902326}{1901089428840953995} a^{17} + \frac{85596927}{271584204120136285} a^{15} + \frac{2505305169}{38797743445733755} a^{13} - \frac{25812550767}{5542534777961965} a^{11} - \frac{14201857592}{158358136513199} a^{9} - \frac{24515505233}{22622590930457} a^{7} - \frac{16043783557}{3231798704351} a^{5} - \frac{22099821665}{461685529193} a^{3} - \frac{23152528759}{65955075599} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{44}\times C_{30844}$, which has order $347426816$ (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:  \( 27849361.142223846 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-210}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{3}, \sqrt{-70})\), 5.5.390625.1, 10.0.102102525000000000000000.1, 10.10.37968750000000000.1, 10.0.420175000000000000000.3

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 R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
3Data not computed
5Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$