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Properties

Label 20.0.34403484087...0041.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{18}\cdot 11^{8}\cdot 23^{10}$
Root discriminant $33.64$
Ramified primes $3, 11, 23$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![351, 216, 1647, 1980, 2853, 1458, -942, -2010, -1254, 522, 1575, 573, 8, -416, -47, 74, 17, 9, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 + 9*x^17 + 17*x^16 + 74*x^15 - 47*x^14 - 416*x^13 + 8*x^12 + 573*x^11 + 1575*x^10 + 522*x^9 - 1254*x^8 - 2010*x^7 - 942*x^6 + 1458*x^5 + 2853*x^4 + 1980*x^3 + 1647*x^2 + 216*x + 351)
 
gp: K = bnfinit(x^20 - x^19 + 3*x^18 + 9*x^17 + 17*x^16 + 74*x^15 - 47*x^14 - 416*x^13 + 8*x^12 + 573*x^11 + 1575*x^10 + 522*x^9 - 1254*x^8 - 2010*x^7 - 942*x^6 + 1458*x^5 + 2853*x^4 + 1980*x^3 + 1647*x^2 + 216*x + 351, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 3 x^{18} + 9 x^{17} + 17 x^{16} + 74 x^{15} - 47 x^{14} - 416 x^{13} + 8 x^{12} + 573 x^{11} + 1575 x^{10} + 522 x^{9} - 1254 x^{8} - 2010 x^{7} - 942 x^{6} + 1458 x^{5} + 2853 x^{4} + 1980 x^{3} + 1647 x^{2} + 216 x + 351 \)

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:  \(3440348408794801675464162130041=3^{18}\cdot 11^{8}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.64$
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:  $3, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6}$, $\frac{1}{99} a^{17} - \frac{7}{99} a^{16} - \frac{1}{33} a^{14} + \frac{47}{99} a^{13} + \frac{26}{99} a^{12} + \frac{49}{99} a^{11} - \frac{23}{99} a^{10} + \frac{32}{99} a^{9} + \frac{5}{11} a^{8} - \frac{13}{33} a^{7} + \frac{16}{33} a^{6} - \frac{16}{33} a^{5} - \frac{13}{33} a^{4} + \frac{4}{33} a^{3} + \frac{3}{11} a^{2} - \frac{5}{11} a - \frac{2}{11}$, $\frac{1}{1881} a^{18} + \frac{1}{1881} a^{17} + \frac{241}{1881} a^{16} + \frac{65}{627} a^{15} + \frac{188}{1881} a^{14} - \frac{185}{627} a^{13} - \frac{337}{1881} a^{12} - \frac{36}{209} a^{11} + \frac{673}{1881} a^{10} + \frac{334}{1881} a^{9} + \frac{98}{209} a^{8} - \frac{5}{19} a^{7} + \frac{85}{209} a^{6} - \frac{69}{209} a^{5} - \frac{265}{627} a^{4} - \frac{256}{627} a^{3} - \frac{3}{209} a^{2} + \frac{3}{11} a + \frac{83}{209}$, $\frac{1}{41446583984684539028245583731509} a^{19} + \frac{7538517582185786430187809380}{41446583984684539028245583731509} a^{18} + \frac{62249878856971100956467758102}{13815527994894846342748527910503} a^{17} - \frac{1569200541706665331278549162334}{13815527994894846342748527910503} a^{16} + \frac{1190831165670837406641631297781}{41446583984684539028245583731509} a^{15} - \frac{4697259306684888002803741740817}{41446583984684539028245583731509} a^{14} + \frac{12966853151896945081598403788038}{41446583984684539028245583731509} a^{13} - \frac{9464425328323115483215497963506}{41446583984684539028245583731509} a^{12} + \frac{6051648502428481869450070967069}{41446583984684539028245583731509} a^{11} + \frac{281393957296281910685862325127}{1255957090444986031158957082773} a^{10} - \frac{1924126432499456459012151833515}{13815527994894846342748527910503} a^{9} - \frac{2720844995134640359506091832914}{13815527994894846342748527910503} a^{8} + \frac{314817681991044773907522347591}{727133052362886649618343574237} a^{7} + \frac{2261664737015520184160717632493}{13815527994894846342748527910503} a^{6} - \frac{430195166873090235947951714413}{1255957090444986031158957082773} a^{5} + \frac{434169459219297908555642709475}{4605175998298282114249509303501} a^{4} + \frac{464543841903505819890568287331}{1535058666099427371416503101167} a^{3} - \frac{407739365852707880569084878853}{4605175998298282114249509303501} a^{2} - \frac{152665643008131706428238855839}{1535058666099427371416503101167} a - \frac{300321909641556195741410975020}{1535058666099427371416503101167}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 8168530.39139 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{-23}) \), 5.5.5184729.1, 10.4.80644244410323.1, 10.0.618272540479143.1, 10.6.1854817621437429.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.6.9.4$x^{6} + 6 x^{4} + 6$$6$$1$$9$$D_{6}$$[2]_{2}^{2}$
3.6.9.4$x^{6} + 6 x^{4} + 6$$6$$1$$9$$D_{6}$$[2]_{2}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$