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Properties

Label 20.0.24527214271...2256.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{18}\cdot 29^{10}$
Root discriminant $93.21$
Ramified primes $2, 11, 29$
Class number $998250$ (GRH)
Class group $[11, 55, 1650]$ (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![4390785169, 0, 4048358601, 0, 1670980114, 0, 405259009, 0, 63737356, 0, 6788948, 0, 502866, 0, 27204, 0, 1200, 0, 45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 45*x^18 + 1200*x^16 + 27204*x^14 + 502866*x^12 + 6788948*x^10 + 63737356*x^8 + 405259009*x^6 + 1670980114*x^4 + 4048358601*x^2 + 4390785169)
 
gp: K = bnfinit(x^20 + 45*x^18 + 1200*x^16 + 27204*x^14 + 502866*x^12 + 6788948*x^10 + 63737356*x^8 + 405259009*x^6 + 1670980114*x^4 + 4048358601*x^2 + 4390785169, 1)
 

Normalized defining polynomial

\( x^{20} + 45 x^{18} + 1200 x^{16} + 27204 x^{14} + 502866 x^{12} + 6788948 x^{10} + 63737356 x^{8} + 405259009 x^{6} + 1670980114 x^{4} + 4048358601 x^{2} + 4390785169 \)

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:  \(2452721427113262984247045999377966432256=2^{20}\cdot 11^{18}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.21$
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:  $2, 11, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1276=2^{2}\cdot 11\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1276}(1,·)$, $\chi_{1276}(581,·)$, $\chi_{1276}(1217,·)$, $\chi_{1276}(523,·)$, $\chi_{1276}(463,·)$, $\chi_{1276}(465,·)$, $\chi_{1276}(1043,·)$, $\chi_{1276}(57,·)$, $\chi_{1276}(985,·)$, $\chi_{1276}(987,·)$, $\chi_{1276}(927,·)$, $\chi_{1276}(929,·)$, $\chi_{1276}(871,·)$, $\chi_{1276}(1159,·)$, $\chi_{1276}(173,·)$, $\chi_{1276}(175,·)$, $\chi_{1276}(115,·)$, $\chi_{1276}(755,·)$, $\chi_{1276}(697,·)$, $\chi_{1276}(637,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{23} a^{13} + \frac{7}{23} a^{11} - \frac{2}{23} a^{9} + \frac{3}{23} a^{7} - \frac{10}{23} a^{5} + \frac{6}{23} a^{3} + \frac{4}{23} a$, $\frac{1}{23} a^{14} + \frac{7}{23} a^{12} - \frac{2}{23} a^{10} + \frac{3}{23} a^{8} - \frac{10}{23} a^{6} + \frac{6}{23} a^{4} + \frac{4}{23} a^{2}$, $\frac{1}{23} a^{15} - \frac{5}{23} a^{11} - \frac{6}{23} a^{9} - \frac{8}{23} a^{7} + \frac{7}{23} a^{5} + \frac{8}{23} a^{3} - \frac{5}{23} a$, $\frac{1}{23} a^{16} - \frac{5}{23} a^{12} - \frac{6}{23} a^{10} - \frac{8}{23} a^{8} + \frac{7}{23} a^{6} + \frac{8}{23} a^{4} - \frac{5}{23} a^{2}$, $\frac{1}{989} a^{17} - \frac{6}{989} a^{15} + \frac{1}{989} a^{13} + \frac{20}{989} a^{11} - \frac{7}{989} a^{9} - \frac{180}{989} a^{7} + \frac{412}{989} a^{5} - \frac{109}{989} a^{3} + \frac{169}{989} a$, $\frac{1}{7851584364930626674236070849} a^{18} - \frac{157169494572545734131121157}{7851584364930626674236070849} a^{16} - \frac{102830302797279417131875241}{7851584364930626674236070849} a^{14} - \frac{648969128237653398221499568}{7851584364930626674236070849} a^{12} + \frac{1610465389074409850540062982}{7851584364930626674236070849} a^{10} - \frac{2470484219719068551957020753}{7851584364930626674236070849} a^{8} + \frac{1035230435123630316966465861}{7851584364930626674236070849} a^{6} - \frac{1520557742802487353436121733}{7851584364930626674236070849} a^{4} - \frac{1911515080531662434230212931}{7851584364930626674236070849} a^{2} - \frac{3882224529773289374737360}{7938912401345426364242741}$, $\frac{1}{526056152450351987173816746883} a^{19} + \frac{231837213093380157716773152}{526056152450351987173816746883} a^{17} + \frac{11218058781521298578278273425}{526056152450351987173816746883} a^{15} + \frac{7932995177616752501524903453}{526056152450351987173816746883} a^{13} - \frac{135010278125679032451713266887}{526056152450351987173816746883} a^{11} - \frac{221282787825601710003215448195}{526056152450351987173816746883} a^{9} + \frac{89752576520158769937379096536}{526056152450351987173816746883} a^{7} + \frac{211663056970526246805574202340}{526056152450351987173816746883} a^{5} - \frac{66502506377878051333709153707}{526056152450351987173816746883} a^{3} - \frac{20709708912804814215631073665}{526056152450351987173816746883} a$

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

Class group and class number

$C_{11}\times C_{55}\times C_{1650}$, which has order $998250$ (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:  \( 281202.490766 \) (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

\(\Q(\sqrt{-319}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-29}) \), \(\Q(\sqrt{11}, \sqrt{-29})\), \(\Q(\zeta_{11})^+\), 10.0.48364216424306959.3, \(\Q(\zeta_{44})^+\), 10.0.4502268874408211456.3

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[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}$
29Data not computed