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Properties

Label 20.0.78487737921...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{18}\cdot 13^{10}$
Root discriminant $139.55$
Ramified primes $2, 5, 11, 13$
Class number $43173680$ (GRH)
Class group $[2, 21586840]$ (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![4762385279521, 0, 2230354203015, 0, 471001418299, 0, 59271734905, 0, 4957175974, 0, 291632342, 0, 12476160, 0, 393684, 0, 8985, 0, 135, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 135*x^18 + 8985*x^16 + 393684*x^14 + 12476160*x^12 + 291632342*x^10 + 4957175974*x^8 + 59271734905*x^6 + 471001418299*x^4 + 2230354203015*x^2 + 4762385279521)
 
gp: K = bnfinit(x^20 + 135*x^18 + 8985*x^16 + 393684*x^14 + 12476160*x^12 + 291632342*x^10 + 4957175974*x^8 + 59271734905*x^6 + 471001418299*x^4 + 2230354203015*x^2 + 4762385279521, 1)
 

Normalized defining polynomial

\( x^{20} + 135 x^{18} + 8985 x^{16} + 393684 x^{14} + 12476160 x^{12} + 291632342 x^{10} + 4957175974 x^{8} + 59271734905 x^{6} + 471001418299 x^{4} + 2230354203015 x^{2} + 4762385279521 \)

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:  \(7848773792186191578606179311298560000000000=2^{20}\cdot 5^{10}\cdot 11^{18}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $139.55$
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:  $2, 5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2860=2^{2}\cdot 5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2860}(1,·)$, $\chi_{2860}(131,·)$, $\chi_{2860}(391,·)$, $\chi_{2860}(521,·)$, $\chi_{2860}(651,·)$, $\chi_{2860}(909,·)$, $\chi_{2860}(1039,·)$, $\chi_{2860}(1299,·)$, $\chi_{2860}(1301,·)$, $\chi_{2860}(1689,·)$, $\chi_{2860}(1691,·)$, $\chi_{2860}(1949,·)$, $\chi_{2860}(1819,·)$, $\chi_{2860}(2341,·)$, $\chi_{2860}(2599,·)$, $\chi_{2860}(129,·)$, $\chi_{2860}(2601,·)$, $\chi_{2860}(2471,·)$, $\chi_{2860}(779,·)$, $\chi_{2860}(1429,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{8501486473992484047765221018716880641} a^{18} - \frac{292270448023843479406906021731905504}{8501486473992484047765221018716880641} a^{16} + \frac{470687180447642117612137577983158124}{8501486473992484047765221018716880641} a^{14} + \frac{3104207345527896880565010513416541090}{8501486473992484047765221018716880641} a^{12} + \frac{367102840025869815218597516948573692}{8501486473992484047765221018716880641} a^{10} - \frac{1832174940263211500770258133503864940}{8501486473992484047765221018716880641} a^{8} - \frac{1911325521407734953414421987256930876}{8501486473992484047765221018716880641} a^{6} + \frac{2426035257864116541061361936728738384}{8501486473992484047765221018716880641} a^{4} + \frac{2201100179078002025799340275991448115}{8501486473992484047765221018716880641} a^{2} - \frac{3260019493407479814106419099862151460}{8501486473992484047765221018716880641}$, $\frac{1}{18552700415842584020113516411714642737167249} a^{19} + \frac{4431135986217440519048661327248572078915875}{18552700415842584020113516411714642737167249} a^{17} + \frac{884274084793034683386369311178395606151098}{18552700415842584020113516411714642737167249} a^{15} - \frac{3098652691779331027768778252775490107013154}{18552700415842584020113516411714642737167249} a^{13} - \frac{8115263576376165471605431809239055804905678}{18552700415842584020113516411714642737167249} a^{11} - \frac{7766135230626496418297173466519060120060363}{18552700415842584020113516411714642737167249} a^{9} - \frac{2630837909621339551220187649990038272572172}{18552700415842584020113516411714642737167249} a^{7} + \frac{6503452545937080326007884305817938647729282}{18552700415842584020113516411714642737167249} a^{5} - \frac{6790934185406173224448546159084655963298839}{18552700415842584020113516411714642737167249} a^{3} - \frac{6049916079506764830390978339968593632706290}{18552700415842584020113516411714642737167249} a$

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

Class group and class number

$C_{2}\times C_{21586840}$, which has order $43173680$ (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:  \( 281202.4907663525 \) (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

\(\Q(\sqrt{-715}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{11}, \sqrt{-65})\), \(\Q(\zeta_{11})^+\), 10.0.2735904600107696875.1, \(\Q(\zeta_{44})^+\), 10.0.254687846410025600000.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
13Data not computed