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Properties

Label 20.0.16863534401...0000.8
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $204.81$
Ramified primes $2, 5, 7, 11$
Class number $41801632$ (GRH)
Class group $[2, 4, 5225204]$ (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![149429406721, 0, 28830230985080, 0, 13317837505399, 0, 2359546225988, 0, 209130391771, 0, 10503433808, 0, 317172100, 0, 5859812, 0, 64729, 0, 392, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 392*x^18 + 64729*x^16 + 5859812*x^14 + 317172100*x^12 + 10503433808*x^10 + 209130391771*x^8 + 2359546225988*x^6 + 13317837505399*x^4 + 28830230985080*x^2 + 149429406721)
 
gp: K = bnfinit(x^20 + 392*x^18 + 64729*x^16 + 5859812*x^14 + 317172100*x^12 + 10503433808*x^10 + 209130391771*x^8 + 2359546225988*x^6 + 13317837505399*x^4 + 28830230985080*x^2 + 149429406721, 1)
 

Normalized defining polynomial

\( x^{20} + 392 x^{18} + 64729 x^{16} + 5859812 x^{14} + 317172100 x^{12} + 10503433808 x^{10} + 209130391771 x^{8} + 2359546225988 x^{6} + 13317837505399 x^{4} + 28830230985080 x^{2} + 149429406721 \)

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:  \(16863534401027144382603387201975746560000000000=2^{40}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $204.81$
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:  $2, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3080=2^{3}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3080}(1,·)$, $\chi_{3080}(1091,·)$, $\chi_{3080}(69,·)$, $\chi_{3080}(519,·)$, $\chi_{3080}(841,·)$, $\chi_{3080}(1931,·)$, $\chi_{3080}(1359,·)$, $\chi_{3080}(1681,·)$, $\chi_{3080}(2771,·)$, $\chi_{3080}(2199,·)$, $\chi_{3080}(1371,·)$, $\chi_{3080}(2589,·)$, $\chi_{3080}(799,·)$, $\chi_{3080}(1189,·)$, $\chi_{3080}(1961,·)$, $\chi_{3080}(811,·)$, $\chi_{3080}(2029,·)$, $\chi_{3080}(239,·)$, $\chi_{3080}(2869,·)$, $\chi_{3080}(1401,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{49} a^{4}$, $\frac{1}{49} a^{5}$, $\frac{1}{343} a^{6}$, $\frac{1}{343} a^{7}$, $\frac{1}{2401} a^{8}$, $\frac{1}{2401} a^{9}$, $\frac{1}{16807} a^{10}$, $\frac{1}{386561} a^{11} + \frac{10}{55223} a^{9} + \frac{5}{7889} a^{7} + \frac{9}{1127} a^{5} - \frac{1}{23} a^{3} + \frac{5}{23} a$, $\frac{1}{1604614711} a^{12} - \frac{4705}{229230673} a^{10} - \frac{5354}{32747239} a^{8} - \frac{3740}{4678177} a^{6} + \frac{4846}{668311} a^{4} - \frac{3560}{95473} a^{2} + \frac{101}{593}$, $\frac{1}{1604614711} a^{13} + \frac{39}{229230673} a^{11} + \frac{167}{4678177} a^{9} + \frac{6341}{4678177} a^{7} + \frac{6625}{668311} a^{5} + \frac{4149}{95473} a^{3} - \frac{1235}{13639} a$, $\frac{1}{11232302977} a^{14} - \frac{6282}{229230673} a^{10} - \frac{3077}{32747239} a^{8} + \frac{2456}{4678177} a^{6} + \frac{6101}{668311} a^{4} + \frac{1215}{95473} a^{2} + \frac{212}{593}$, $\frac{1}{11232302977} a^{15} + \frac{241}{229230673} a^{11} - \frac{6042}{32747239} a^{9} - \frac{5846}{4678177} a^{7} - \frac{3387}{668311} a^{5} - \frac{3529}{95473} a^{3} - \frac{3426}{13639} a$, $\frac{1}{78626120839} a^{16} - \frac{4174}{229230673} a^{10} + \frac{2402}{32747239} a^{8} - \frac{2221}{4678177} a^{6} + \frac{1539}{668311} a^{4} - \frac{4723}{95473} a^{2} - \frac{28}{593}$, $\frac{1}{78626120839} a^{17} - \frac{1}{9966551} a^{11} + \frac{2995}{32747239} a^{9} + \frac{4895}{4678177} a^{7} - \frac{2019}{668311} a^{5} - \frac{6502}{95473} a^{3} + \frac{6472}{13639} a$, $\frac{1}{550382845873} a^{18} + \frac{556}{32747239} a^{10} + \frac{4504}{32747239} a^{8} - \frac{6205}{4678177} a^{6} - \frac{4156}{668311} a^{4} + \frac{918}{13639} a^{2} - \frac{49}{593}$, $\frac{1}{550382845873} a^{19} - \frac{37}{32747239} a^{11} + \frac{3911}{32747239} a^{9} + \frac{318}{4678177} a^{7} - \frac{26}{29057} a^{5} - \frac{5434}{95473} a^{3} + \frac{5396}{13639} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{5225204}$, which has order $41801632$ (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:  \( 11184526.893275889 \) (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

\(\Q(\sqrt{-154}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{55}, \sqrt{-70})\), \(\Q(\zeta_{11})^+\), 10.0.1298596719579529216.1, 10.0.368919522607820800000.1, 10.10.7545432611200000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
7Data not computed
11Data not computed