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Properties

Label 20.12.2572121384...0000.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $37.20$
Ramified primes $2, 5, 3469$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-124, -628, 920, 6556, 164, -16856, -8568, 7624, -3112, -12796, -956, 7146, 2054, -1692, -758, 182, 158, -6, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 - 6*x^17 + 158*x^16 + 182*x^15 - 758*x^14 - 1692*x^13 + 2054*x^12 + 7146*x^11 - 956*x^10 - 12796*x^9 - 3112*x^8 + 7624*x^7 - 8568*x^6 - 16856*x^5 + 164*x^4 + 6556*x^3 + 920*x^2 - 628*x - 124)
 
gp: K = bnfinit(x^20 - 20*x^18 - 6*x^17 + 158*x^16 + 182*x^15 - 758*x^14 - 1692*x^13 + 2054*x^12 + 7146*x^11 - 956*x^10 - 12796*x^9 - 3112*x^8 + 7624*x^7 - 8568*x^6 - 16856*x^5 + 164*x^4 + 6556*x^3 + 920*x^2 - 628*x - 124, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{18} - 6 x^{17} + 158 x^{16} + 182 x^{15} - 758 x^{14} - 1692 x^{13} + 2054 x^{12} + 7146 x^{11} - 956 x^{10} - 12796 x^{9} - 3112 x^{8} + 7624 x^{7} - 8568 x^{6} - 16856 x^{5} + 164 x^{4} + 6556 x^{3} + 920 x^{2} - 628 x - 124 \)

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:  \(25721213848857003468800000000000=2^{20}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.20$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{799910} a^{18} - \frac{98767}{399955} a^{17} - \frac{55973}{799910} a^{16} + \frac{100027}{799910} a^{15} + \frac{112027}{799910} a^{14} + \frac{99408}{399955} a^{13} - \frac{37837}{159982} a^{12} - \frac{20603}{399955} a^{11} + \frac{166873}{799910} a^{10} - \frac{153941}{399955} a^{9} - \frac{2029}{79991} a^{8} + \frac{81296}{399955} a^{7} - \frac{7461}{79991} a^{6} + \frac{161833}{399955} a^{5} - \frac{124246}{399955} a^{4} + \frac{46834}{399955} a^{3} - \frac{26890}{79991} a^{2} - \frac{22808}{399955} a + \frac{83727}{399955}$, $\frac{1}{3162980719013630830491683016340} a^{19} + \frac{85610865520109962079718}{790745179753407707622920754085} a^{18} - \frac{155471286000645816060965386128}{790745179753407707622920754085} a^{17} + \frac{144860417387860164104908693881}{790745179753407707622920754085} a^{16} + \frac{1771491064861404848974744286}{19286467798863602624949286685} a^{15} - \frac{152455076924411094483141981161}{1581490359506815415245841508170} a^{14} + \frac{263134154213580155569379988223}{1581490359506815415245841508170} a^{13} + \frac{18168241821200433804419480657}{1581490359506815415245841508170} a^{12} - \frac{70836676621618200763467848802}{790745179753407707622920754085} a^{11} - \frac{57781924842634236290273563077}{1581490359506815415245841508170} a^{10} - \frac{169840043019538677147960578421}{1581490359506815415245841508170} a^{9} + \frac{68168755867951863716329336013}{790745179753407707622920754085} a^{8} + \frac{89786692979473924948422186053}{790745179753407707622920754085} a^{7} - \frac{30560132159546815043854923896}{790745179753407707622920754085} a^{6} + \frac{129920987211501278389896705276}{790745179753407707622920754085} a^{5} - \frac{62797866489440086496393256701}{790745179753407707622920754085} a^{4} + \frac{116382367037665633326153911587}{790745179753407707622920754085} a^{3} - \frac{253256773508123539845235428224}{790745179753407707622920754085} a^{2} - \frac{323752817475647222440975675473}{790745179753407707622920754085} a - \frac{161814024811524333597759670254}{790745179753407707622920754085}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 181753707.715 \) (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 $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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