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Properties

Label 20.0.37990573067...0784.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 29^{12}$
Root discriminant $21.33$
Ramified primes $2, 29$
Class number $1$
Class group Trivial
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![2, -4, 6, -50, 225, -550, 1078, -1830, 2856, -4092, 5242, -5814, 5465, -4298, 2800, -1494, 643, -218, 56, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 56*x^18 - 218*x^17 + 643*x^16 - 1494*x^15 + 2800*x^14 - 4298*x^13 + 5465*x^12 - 5814*x^11 + 5242*x^10 - 4092*x^9 + 2856*x^8 - 1830*x^7 + 1078*x^6 - 550*x^5 + 225*x^4 - 50*x^3 + 6*x^2 - 4*x + 2)
 
gp: K = bnfinit(x^20 - 10*x^19 + 56*x^18 - 218*x^17 + 643*x^16 - 1494*x^15 + 2800*x^14 - 4298*x^13 + 5465*x^12 - 5814*x^11 + 5242*x^10 - 4092*x^9 + 2856*x^8 - 1830*x^7 + 1078*x^6 - 550*x^5 + 225*x^4 - 50*x^3 + 6*x^2 - 4*x + 2, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 56 x^{18} - 218 x^{17} + 643 x^{16} - 1494 x^{15} + 2800 x^{14} - 4298 x^{13} + 5465 x^{12} - 5814 x^{11} + 5242 x^{10} - 4092 x^{9} + 2856 x^{8} - 1830 x^{7} + 1078 x^{6} - 550 x^{5} + 225 x^{4} - 50 x^{3} + 6 x^{2} - 4 x + 2 \)

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:  \(379905730677204894858870784=2^{30}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.33$
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:  $2, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{219948561351283} a^{19} - \frac{22686025959734}{219948561351283} a^{18} - \frac{97751955177068}{219948561351283} a^{17} - \frac{53435503436273}{219948561351283} a^{16} - \frac{12154714323695}{219948561351283} a^{15} + \frac{45106362014274}{219948561351283} a^{14} + \frac{100050094377592}{219948561351283} a^{13} - \frac{60964935005447}{219948561351283} a^{12} + \frac{104947935368373}{219948561351283} a^{11} + \frac{28189681017201}{219948561351283} a^{10} + \frac{38709215740954}{219948561351283} a^{9} + \frac{31778652240504}{219948561351283} a^{8} - \frac{79895294802220}{219948561351283} a^{7} - \frac{11963119858033}{219948561351283} a^{6} + \frac{96137559991180}{219948561351283} a^{5} - \frac{38747793102914}{219948561351283} a^{4} - \frac{25233004283929}{219948561351283} a^{3} - \frac{2002197306216}{219948561351283} a^{2} + \frac{71758353006484}{219948561351283} a + \frac{83320092289321}{219948561351283}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 447869.030966 \)
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

5.1.53824.1 x2, 10.2.2436396322816.1

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

Sibling fields

Degree 5 sibling: 5.1.53824.1
Degree 6 sibling: 6.2.45265984.4
Degree 10 sibling: 10.2.2436396322816.1
Degree 12 sibling: 12.0.131136595679248384.6
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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.12.18.59$x^{12} - 2 x^{11} + 6 x^{10} + 4 x^{9} + 6 x^{8} + 12 x^{7} - 4 x^{6} - 8 x^{3} + 16 x^{2} - 8$$4$$3$$18$$A_4$$[2, 2]^{3}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$