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Properties

Label 20.16.3956448137...6016.2
Degree $20$
Signature $[16, 2]$
Discriminant $2^{40}\cdot 11^{16}\cdot 23^{8}$
Root discriminant $95.47$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T331

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![541696, 0, -4333568, 0, 10698496, 0, -10103808, 0, 3090432, 0, -126848, 0, -69680, 0, 9328, 0, -188, 0, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 24*x^18 - 188*x^16 + 9328*x^14 - 69680*x^12 - 126848*x^10 + 3090432*x^8 - 10103808*x^6 + 10698496*x^4 - 4333568*x^2 + 541696)
 
gp: K = bnfinit(x^20 - 24*x^18 - 188*x^16 + 9328*x^14 - 69680*x^12 - 126848*x^10 + 3090432*x^8 - 10103808*x^6 + 10698496*x^4 - 4333568*x^2 + 541696, 1)
 

Normalized defining polynomial

\( x^{20} - 24 x^{18} - 188 x^{16} + 9328 x^{14} - 69680 x^{12} - 126848 x^{10} + 3090432 x^{8} - 10103808 x^{6} + 10698496 x^{4} - 4333568 x^{2} + 541696 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3956448137628099441486726576701509206016=2^{40}\cdot 11^{16}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.47$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{64} a^{11} - \frac{1}{32} a^{9} - \frac{1}{16} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{12}$, $\frac{1}{128} a^{13} - \frac{1}{16} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a$, $\frac{1}{2944} a^{14} + \frac{7}{1472} a^{12} - \frac{3}{368} a^{10} - \frac{3}{184} a^{8} - \frac{7}{184} a^{6} - \frac{3}{92} a^{4}$, $\frac{1}{5888} a^{15} + \frac{7}{2944} a^{13} - \frac{3}{736} a^{11} + \frac{17}{736} a^{9} + \frac{1}{23} a^{7} - \frac{3}{184} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{641792} a^{16} - \frac{9}{80224} a^{14} + \frac{1027}{160448} a^{12} + \frac{459}{80224} a^{10} - \frac{185}{20056} a^{8} + \frac{45}{20056} a^{6} + \frac{559}{5014} a^{4} - \frac{41}{218} a^{2} - \frac{38}{109}$, $\frac{1}{1283584} a^{17} - \frac{9}{160448} a^{15} + \frac{1027}{320896} a^{13} + \frac{459}{160448} a^{11} + \frac{2137}{80224} a^{9} + \frac{45}{40112} a^{7} + \frac{559}{10028} a^{5} + \frac{17}{109} a^{3} + \frac{71}{218} a$, $\frac{1}{3043045399296512} a^{18} - \frac{51972421}{1521522699648256} a^{16} + \frac{63560772237}{760761349824128} a^{14} + \frac{2317905019607}{380380674912064} a^{12} + \frac{159522650253}{95095168728016} a^{10} + \frac{1136506093793}{47547584364008} a^{8} + \frac{1015340937777}{23773792182004} a^{6} + \frac{303907716271}{11886896091002} a^{4} - \frac{16763281763}{258410784587} a^{2} - \frac{102463329065}{258410784587}$, $\frac{1}{6086090798593024} a^{19} - \frac{51972421}{3043045399296512} a^{17} + \frac{63560772237}{1521522699648256} a^{15} + \frac{2317905019607}{760761349824128} a^{13} + \frac{159522650253}{190190337456032} a^{11} + \frac{1136506093793}{95095168728016} a^{9} + \frac{1015340937777}{47547584364008} a^{7} - \frac{5335632612959}{47547584364008} a^{5} + \frac{224884221061}{1033643138348} a^{3} + \frac{77973727761}{258410784587} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 9580828508890 \) (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

\(\Q(\zeta_{11})^+\), 10.10.2670699013250048.1

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 ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$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}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$