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Properties

Label 20.0.14138200398...4176.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 11^{16}\cdot 23^{4}$
Root discriminant $50.99$
Ramified primes $2, 11, 23$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 20T331

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![279841, 0, -152030, 0, 58444, 0, -8040, 0, -109, 0, -2848, 0, 1, 0, 210, 0, 89, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 12*x^18 + 89*x^16 + 210*x^14 + x^12 - 2848*x^10 - 109*x^8 - 8040*x^6 + 58444*x^4 - 152030*x^2 + 279841)
 
gp: K = bnfinit(x^20 + 12*x^18 + 89*x^16 + 210*x^14 + x^12 - 2848*x^10 - 109*x^8 - 8040*x^6 + 58444*x^4 - 152030*x^2 + 279841, 1)
 

Normalized defining polynomial

\( x^{20} + 12 x^{18} + 89 x^{16} + 210 x^{14} + x^{12} - 2848 x^{10} - 109 x^{8} - 8040 x^{6} + 58444 x^{4} - 152030 x^{2} + 279841 \)

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:  \(14138200398183609412083027778994176=2^{40}\cdot 11^{16}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.99$
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:  $2, 11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} - \frac{5}{11} a^{8} - \frac{1}{11} a^{6} + \frac{1}{11} a^{4} + \frac{5}{11} a^{2} - \frac{1}{11}$, $\frac{1}{11} a^{11} - \frac{5}{11} a^{9} - \frac{1}{11} a^{7} + \frac{1}{11} a^{5} + \frac{5}{11} a^{3} - \frac{1}{11} a$, $\frac{1}{11} a^{12} - \frac{4}{11} a^{8} - \frac{4}{11} a^{6} - \frac{1}{11} a^{4} + \frac{2}{11} a^{2} - \frac{5}{11}$, $\frac{1}{11} a^{13} - \frac{4}{11} a^{9} - \frac{4}{11} a^{7} - \frac{1}{11} a^{5} + \frac{2}{11} a^{3} - \frac{5}{11} a$, $\frac{1}{11} a^{14} - \frac{2}{11} a^{8} - \frac{5}{11} a^{6} - \frac{5}{11} a^{4} + \frac{4}{11} a^{2} - \frac{4}{11}$, $\frac{1}{11} a^{15} - \frac{2}{11} a^{9} - \frac{5}{11} a^{7} - \frac{5}{11} a^{5} + \frac{4}{11} a^{3} - \frac{4}{11} a$, $\frac{1}{253} a^{16} + \frac{1}{23} a^{14} + \frac{9}{253} a^{12} - \frac{6}{253} a^{10} + \frac{122}{253} a^{8} - \frac{26}{253} a^{6} + \frac{101}{253} a^{4} + \frac{93}{253} a^{2} + \frac{3}{11}$, $\frac{1}{253} a^{17} + \frac{1}{23} a^{15} + \frac{9}{253} a^{13} - \frac{6}{253} a^{11} + \frac{122}{253} a^{9} - \frac{26}{253} a^{7} + \frac{101}{253} a^{5} + \frac{93}{253} a^{3} + \frac{3}{11} a$, $\frac{1}{29956633888419888089} a^{18} - \frac{7961407116289743}{29956633888419888089} a^{16} - \frac{476240380641105418}{29956633888419888089} a^{14} - \frac{65831364467632032}{2723330353492717099} a^{12} + \frac{606078885135554482}{29956633888419888089} a^{10} - \frac{8254732628687561656}{29956633888419888089} a^{8} + \frac{308348033155128014}{1302462342974777743} a^{6} + \frac{14692681661743371974}{29956633888419888089} a^{4} + \frac{14663919454168574896}{29956633888419888089} a^{2} - \frac{66490181144049230}{1302462342974777743}$, $\frac{1}{689002579433657426047} a^{19} - \frac{836801079918421034}{689002579433657426047} a^{17} - \frac{129407792816945302}{62636598130332493277} a^{15} + \frac{2709619349607734425}{689002579433657426047} a^{13} + \frac{21919099042904644822}{689002579433657426047} a^{11} + \frac{198363157234129453029}{689002579433657426047} a^{9} + \frac{287358219837231482293}{689002579433657426047} a^{7} + \frac{4123613595947475431}{62636598130332493277} a^{5} + \frac{193574883111885772139}{689002579433657426047} a^{3} - \frac{14867198624039250855}{29956633888419888089} a$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 151756154.41 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T331:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 56 conjugacy class representatives for t20n331 are not computed
Character table for t20n331 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.0.5048580365312.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$