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Properties

Label 20.12.6584630745...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{28}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $49.08$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-20956, -353496, 504812, 575880, -886104, 118692, -6948, 97080, 24668, -56488, 31022, -10284, -370, 1708, -1630, 838, -88, -56, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 28*x^18 - 56*x^17 - 88*x^16 + 838*x^15 - 1630*x^14 + 1708*x^13 - 370*x^12 - 10284*x^11 + 31022*x^10 - 56488*x^9 + 24668*x^8 + 97080*x^7 - 6948*x^6 + 118692*x^5 - 886104*x^4 + 575880*x^3 + 504812*x^2 - 353496*x - 20956)
 
gp: K = bnfinit(x^20 - 8*x^19 + 28*x^18 - 56*x^17 - 88*x^16 + 838*x^15 - 1630*x^14 + 1708*x^13 - 370*x^12 - 10284*x^11 + 31022*x^10 - 56488*x^9 + 24668*x^8 + 97080*x^7 - 6948*x^6 + 118692*x^5 - 886104*x^4 + 575880*x^3 + 504812*x^2 - 353496*x - 20956, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 28 x^{18} - 56 x^{17} - 88 x^{16} + 838 x^{15} - 1630 x^{14} + 1708 x^{13} - 370 x^{12} - 10284 x^{11} + 31022 x^{10} - 56488 x^{9} + 24668 x^{8} + 97080 x^{7} - 6948 x^{6} + 118692 x^{5} - 886104 x^{4} + 575880 x^{3} + 504812 x^{2} - 353496 x - 20956 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6584630745307392888012800000000000=2^{28}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.08$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{2} a^{18}$, $\frac{1}{14806677953052088608398717374864206907441399122493025086} a^{19} - \frac{2041780578024701928540024904257712592609992503876839713}{14806677953052088608398717374864206907441399122493025086} a^{18} + \frac{1034751781483538011992662207136796727740812041390058709}{7403338976526044304199358687432103453720699561246512543} a^{17} + \frac{954604402879545151929000938628788092978512378111274033}{14806677953052088608398717374864206907441399122493025086} a^{16} - \frac{3253516480946335243525151996431447336759581423123164555}{14806677953052088608398717374864206907441399122493025086} a^{15} - \frac{971597217559748091097161636451893086291541657231872023}{7403338976526044304199358687432103453720699561246512543} a^{14} + \frac{948287165411894251086134848908020953847670938758224303}{7403338976526044304199358687432103453720699561246512543} a^{13} + \frac{404530533275855516391309120842098828215396400912811619}{7403338976526044304199358687432103453720699561246512543} a^{12} + \frac{2751570844895564547486131975464979771397338525213772497}{14806677953052088608398717374864206907441399122493025086} a^{11} + \frac{2210828039991544313165901204410779913348569335846848285}{14806677953052088608398717374864206907441399122493025086} a^{10} - \frac{604631886016081427238372780018275302272575694464456829}{7403338976526044304199358687432103453720699561246512543} a^{9} - \frac{3678646514094774854053023029562162578536002796875296794}{7403338976526044304199358687432103453720699561246512543} a^{8} - \frac{1005971609642704202908063366744626476970966975582530074}{7403338976526044304199358687432103453720699561246512543} a^{7} + \frac{1477545754655669017349055648533262190475651054701754949}{7403338976526044304199358687432103453720699561246512543} a^{6} - \frac{3120752437185216758401776867781837867160336791271721877}{7403338976526044304199358687432103453720699561246512543} a^{5} + \frac{1520141851175033167620234488234235993013756522225055776}{7403338976526044304199358687432103453720699561246512543} a^{4} + \frac{904292912841157391245955431282805974783129939336391159}{7403338976526044304199358687432103453720699561246512543} a^{3} - \frac{687359202227462761730540630257716273579947009397152884}{7403338976526044304199358687432103453720699561246512543} a^{2} - \frac{943794150876262283020976342609326045232538927810083722}{7403338976526044304199358687432103453720699561246512543} a - \frac{250826887633935916638669795985313986189999672507253149}{569487613578926484938412206725546419516976889326654811}$

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:  $15$
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:  \( 6670883258.68 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed