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Properties

Label 20.20.3362222060...5625.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{26}\cdot 41^{12}$
Root discriminant $75.22$
Ramified primes $5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $A_5$ (as 20T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-115, -4575, -18175, 69475, 436755, 452545, -495745, -860395, 152230, 594910, 16401, -212780, -16315, 43940, 2720, -5333, -85, 355, -15, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 15*x^18 + 355*x^17 - 85*x^16 - 5333*x^15 + 2720*x^14 + 43940*x^13 - 16315*x^12 - 212780*x^11 + 16401*x^10 + 594910*x^9 + 152230*x^8 - 860395*x^7 - 495745*x^6 + 452545*x^5 + 436755*x^4 + 69475*x^3 - 18175*x^2 - 4575*x - 115)
 
gp: K = bnfinit(x^20 - 10*x^19 - 15*x^18 + 355*x^17 - 85*x^16 - 5333*x^15 + 2720*x^14 + 43940*x^13 - 16315*x^12 - 212780*x^11 + 16401*x^10 + 594910*x^9 + 152230*x^8 - 860395*x^7 - 495745*x^6 + 452545*x^5 + 436755*x^4 + 69475*x^3 - 18175*x^2 - 4575*x - 115, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} - 15 x^{18} + 355 x^{17} - 85 x^{16} - 5333 x^{15} + 2720 x^{14} + 43940 x^{13} - 16315 x^{12} - 212780 x^{11} + 16401 x^{10} + 594910 x^{9} + 152230 x^{8} - 860395 x^{7} - 495745 x^{6} + 452545 x^{5} + 436755 x^{4} + 69475 x^{3} - 18175 x^{2} - 4575 x - 115 \)

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:  \(33622220606157460929453372955322265625=5^{26}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.22$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{20} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{5} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{260} a^{16} + \frac{1}{260} a^{15} - \frac{5}{26} a^{14} + \frac{11}{52} a^{12} - \frac{1}{65} a^{11} + \frac{11}{260} a^{10} - \frac{9}{26} a^{9} - \frac{3}{26} a^{8} - \frac{9}{26} a^{7} - \frac{11}{26} a^{6} - \frac{3}{52} a^{5} + \frac{23}{52} a^{4} - \frac{1}{26} a^{3} - \frac{1}{13} a^{2} + \frac{5}{52} a - \frac{1}{13}$, $\frac{1}{1300} a^{17} + \frac{1}{650} a^{16} + \frac{4}{325} a^{15} + \frac{3}{260} a^{14} + \frac{63}{260} a^{13} - \frac{79}{1300} a^{12} - \frac{47}{325} a^{11} - \frac{209}{1300} a^{10} + \frac{119}{260} a^{9} + \frac{27}{130} a^{8} - \frac{2}{13} a^{7} + \frac{1}{260} a^{6} - \frac{3}{130} a^{5} - \frac{11}{65} a^{4} + \frac{17}{52} a^{3} + \frac{27}{260} a^{2} + \frac{79}{260} a - \frac{43}{260}$, $\frac{1}{287300} a^{18} + \frac{67}{287300} a^{17} + \frac{331}{287300} a^{16} - \frac{71}{28730} a^{15} + \frac{699}{28730} a^{14} - \frac{31727}{143650} a^{13} + \frac{7388}{71825} a^{12} + \frac{837}{22100} a^{11} - \frac{4661}{57460} a^{10} - \frac{2037}{4420} a^{9} + \frac{53}{2873} a^{8} + \frac{12011}{57460} a^{7} + \frac{1477}{3380} a^{6} - \frac{24129}{57460} a^{5} - \frac{303}{2873} a^{4} + \frac{4671}{28730} a^{3} + \frac{491}{1690} a^{2} - \frac{22843}{57460} a - \frac{5257}{11492}$, $\frac{1}{4225268876385661702850929300} a^{19} + \frac{1261893393408785031859}{2112634438192830851425464650} a^{18} + \frac{264924199867155064793031}{2112634438192830851425464650} a^{17} + \frac{2428684717855807510499559}{4225268876385661702850929300} a^{16} - \frac{79025908487197525937331401}{4225268876385661702850929300} a^{15} - \frac{243243603486574734642596729}{4225268876385661702850929300} a^{14} - \frac{2654871804806904619089219}{35807363359200522905516350} a^{13} - \frac{661924396772076004887506473}{4225268876385661702850929300} a^{12} + \frac{558468960682455219086426429}{4225268876385661702850929300} a^{11} - \frac{205196378432501248093578104}{1056317219096415425712732325} a^{10} + \frac{7031963451231520381790575}{42252688763856617028509293} a^{9} + \frac{10436450852191651187544641}{49709045604537196504128580} a^{8} - \frac{7678884548449017636099025}{42252688763856617028509293} a^{7} - \frac{183919650982179876047532243}{422526887638566170285092930} a^{6} + \frac{196587965703283857671203697}{845053775277132340570185860} a^{5} - \frac{27572011643506791188377213}{65004136559779410813091220} a^{4} - \frac{233871011097024132908146829}{845053775277132340570185860} a^{3} - \frac{319618070951667167327103281}{845053775277132340570185860} a^{2} - \frac{88096487885015196918186783}{211263443819283085142546465} a + \frac{192756265657384263263550929}{422526887638566170285092930}$

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:  $19$
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:  \( 1908142429540 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_5$ (as 20T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 60
The 5 conjugacy class representatives for $A_5$
Character table for $A_5$

Intermediate fields

5.5.26265625.1 x2, 10.10.1159693418212890625.1

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

Sibling fields

Degree 5 sibling: 5.5.26265625.1
Degree 6 sibling: 6.6.44152515625.1
Degree 10 sibling: 10.10.1159693418212890625.1
Degree 12 sibling: Deg 12
Degree 15 sibling: Deg 15
Degree 30 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$5$5.10.13.7$x^{10} + 5 x^{4} + 10$$10$$1$$13$$D_5$$[3/2]_{2}$
5.10.13.7$x^{10} + 5 x^{4} + 10$$10$$1$$13$$D_5$$[3/2]_{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$