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Properties

Label 20.0.33302510381...9401.1
Degree $20$
Signature $[0, 10]$
Discriminant $19^{10}\cdot 293^{8}$
Root discriminant $42.28$
Ramified primes $19, 293$
Class number $14$ (GRH)
Class group $[14]$ (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![77, -9, 3265, 4718, -1363, -14351, -4769, 15281, 15528, 2997, -2385, 655, 1233, 354, -7, -159, 21, -19, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 3*x^18 - 19*x^17 + 21*x^16 - 159*x^15 - 7*x^14 + 354*x^13 + 1233*x^12 + 655*x^11 - 2385*x^10 + 2997*x^9 + 15528*x^8 + 15281*x^7 - 4769*x^6 - 14351*x^5 - 1363*x^4 + 4718*x^3 + 3265*x^2 - 9*x + 77)
 
gp: K = bnfinit(x^20 + 3*x^18 - 19*x^17 + 21*x^16 - 159*x^15 - 7*x^14 + 354*x^13 + 1233*x^12 + 655*x^11 - 2385*x^10 + 2997*x^9 + 15528*x^8 + 15281*x^7 - 4769*x^6 - 14351*x^5 - 1363*x^4 + 4718*x^3 + 3265*x^2 - 9*x + 77, 1)
 

Normalized defining polynomial

\( x^{20} + 3 x^{18} - 19 x^{17} + 21 x^{16} - 159 x^{15} - 7 x^{14} + 354 x^{13} + 1233 x^{12} + 655 x^{11} - 2385 x^{10} + 2997 x^{9} + 15528 x^{8} + 15281 x^{7} - 4769 x^{6} - 14351 x^{5} - 1363 x^{4} + 4718 x^{3} + 3265 x^{2} - 9 x + 77 \)

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:  \(333025103817911062616373171479401=19^{10}\cdot 293^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.28$
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:  $19, 293$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{8742408011955969067472254218552834905522005169} a^{19} + \frac{4228053835032209217275525704531534115915097802}{8742408011955969067472254218552834905522005169} a^{18} + \frac{2394743080548428161536613219084499513620240938}{8742408011955969067472254218552834905522005169} a^{17} - \frac{2748745860769560408150321629848952856337023436}{8742408011955969067472254218552834905522005169} a^{16} - \frac{2871974515423692822049258279217514316726399889}{8742408011955969067472254218552834905522005169} a^{15} + \frac{526766762236013588759243434414938765468897695}{8742408011955969067472254218552834905522005169} a^{14} - \frac{3897542106235888436467162444385257457801159148}{8742408011955969067472254218552834905522005169} a^{13} + \frac{1768226658870234375842753260743085019188086586}{8742408011955969067472254218552834905522005169} a^{12} - \frac{2909977908357368834808279801914065736637728958}{8742408011955969067472254218552834905522005169} a^{11} - \frac{1537170344020972541431170581917596601626180394}{8742408011955969067472254218552834905522005169} a^{10} - \frac{508923673140075547093884281398124715528183315}{8742408011955969067472254218552834905522005169} a^{9} - \frac{536268375715635853428160281840583753594300894}{8742408011955969067472254218552834905522005169} a^{8} - \frac{2713345219336844905669085604920514516210970257}{8742408011955969067472254218552834905522005169} a^{7} - \frac{2925708595169210313571674795348564053088619421}{8742408011955969067472254218552834905522005169} a^{6} - \frac{2919634730828235484651089680047738446846193080}{8742408011955969067472254218552834905522005169} a^{5} - \frac{3715631615934474797438643601235972601956071758}{8742408011955969067472254218552834905522005169} a^{4} - \frac{57192268850892286428902268536969902614277393}{8742408011955969067472254218552834905522005169} a^{3} - \frac{2944085185081970319954326389490632322485241878}{8742408011955969067472254218552834905522005169} a^{2} - \frac{2593483578708076398425297579301425974776329193}{8742408011955969067472254218552834905522005169} a + \frac{358873828360830618244791381233302875100227859}{8742408011955969067472254218552834905522005169}$

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

Class group and class number

$C_{14}$, which has order $14$ (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:  \( 7783721.02563 \) (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

\(\Q(\sqrt{-19}) \), 10.10.960472390437121.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
293Data not computed