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Properties

Label 20.20.5890836881...9249.1
Degree $20$
Signature $[20, 0]$
Discriminant $17^{10}\cdot 643^{8}$
Root discriminant $54.77$
Ramified primes $17, 643$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times A_5$ (as 20T31)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-101, -1660, 7682, 18788, -38499, -61692, 85210, 88980, -107173, -61496, 80377, 17062, -34555, 1448, 7586, -1660, -634, 240, 5, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 5*x^18 + 240*x^17 - 634*x^16 - 1660*x^15 + 7586*x^14 + 1448*x^13 - 34555*x^12 + 17062*x^11 + 80377*x^10 - 61496*x^9 - 107173*x^8 + 88980*x^7 + 85210*x^6 - 61692*x^5 - 38499*x^4 + 18788*x^3 + 7682*x^2 - 1660*x - 101)
 
gp: K = bnfinit(x^20 - 10*x^19 + 5*x^18 + 240*x^17 - 634*x^16 - 1660*x^15 + 7586*x^14 + 1448*x^13 - 34555*x^12 + 17062*x^11 + 80377*x^10 - 61496*x^9 - 107173*x^8 + 88980*x^7 + 85210*x^6 - 61692*x^5 - 38499*x^4 + 18788*x^3 + 7682*x^2 - 1660*x - 101, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 5 x^{18} + 240 x^{17} - 634 x^{16} - 1660 x^{15} + 7586 x^{14} + 1448 x^{13} - 34555 x^{12} + 17062 x^{11} + 80377 x^{10} - 61496 x^{9} - 107173 x^{8} + 88980 x^{7} + 85210 x^{6} - 61692 x^{5} - 38499 x^{4} + 18788 x^{3} + 7682 x^{2} - 1660 x - 101 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(58908368812107185790916140058429249=17^{10}\cdot 643^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.77$
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:  $17, 643$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{9408446} a^{18} - \frac{9}{9408446} a^{17} + \frac{14237}{276719} a^{16} + \frac{48939}{553438} a^{15} + \frac{1135759}{4704223} a^{14} + \frac{116757}{9408446} a^{13} - \frac{5115}{9408446} a^{12} + \frac{1088145}{9408446} a^{11} - \frac{251121}{4704223} a^{10} + \frac{2014723}{9408446} a^{9} - \frac{4631487}{9408446} a^{8} + \frac{4273267}{9408446} a^{7} - \frac{1456826}{4704223} a^{6} - \frac{3479009}{9408446} a^{5} - \frac{2198962}{4704223} a^{4} + \frac{2138798}{4704223} a^{3} - \frac{13503}{276719} a^{2} + \frac{515256}{4704223} a + \frac{2054418}{4704223}$, $\frac{1}{13538753794} a^{19} + \frac{355}{6769376897} a^{18} - \frac{1570732895}{13538753794} a^{17} + \frac{9452985}{796397282} a^{16} + \frac{344916576}{6769376897} a^{15} - \frac{187682511}{6769376897} a^{14} - \frac{2159971203}{13538753794} a^{13} - \frac{928026701}{6769376897} a^{12} - \frac{66355325}{796397282} a^{11} + \frac{123873746}{6769376897} a^{10} - \frac{3248387037}{6769376897} a^{9} + \frac{1488946467}{6769376897} a^{8} + \frac{3169500153}{6769376897} a^{7} - \frac{1089183294}{6769376897} a^{6} + \frac{5453739921}{13538753794} a^{5} - \frac{565802887}{13538753794} a^{4} + \frac{3907779893}{13538753794} a^{3} - \frac{4849822129}{13538753794} a^{2} + \frac{120425110}{398198641} a + \frac{2497942933}{6769376897}$

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:  $19$
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:  \( 94964848909.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_5$ (as 20T31):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 10.10.14277086054271121.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 20 sibling: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
643Data not computed