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Properties

Label 20.0.47931163861...7152.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 53^{15}$
Root discriminant $34.20$
Ramified primes $2, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![848, -8480, 40704, -124656, 272960, -453760, 593720, -626536, 542848, -390744, 234180, -115500, 45612, -13792, 2984, -448, 102, -64, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 30*x^18 - 64*x^17 + 102*x^16 - 448*x^15 + 2984*x^14 - 13792*x^13 + 45612*x^12 - 115500*x^11 + 234180*x^10 - 390744*x^9 + 542848*x^8 - 626536*x^7 + 593720*x^6 - 453760*x^5 + 272960*x^4 - 124656*x^3 + 40704*x^2 - 8480*x + 848)
 
gp: K = bnfinit(x^20 - 8*x^19 + 30*x^18 - 64*x^17 + 102*x^16 - 448*x^15 + 2984*x^14 - 13792*x^13 + 45612*x^12 - 115500*x^11 + 234180*x^10 - 390744*x^9 + 542848*x^8 - 626536*x^7 + 593720*x^6 - 453760*x^5 + 272960*x^4 - 124656*x^3 + 40704*x^2 - 8480*x + 848, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 30 x^{18} - 64 x^{17} + 102 x^{16} - 448 x^{15} + 2984 x^{14} - 13792 x^{13} + 45612 x^{12} - 115500 x^{11} + 234180 x^{10} - 390744 x^{9} + 542848 x^{8} - 626536 x^{7} + 593720 x^{6} - 453760 x^{5} + 272960 x^{4} - 124656 x^{3} + 40704 x^{2} - 8480 x + 848 \)

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:  \(4793116386199379622264055857152=2^{16}\cdot 53^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.20$
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:  $2, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{24} a^{14} - \frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{24} a^{15} - \frac{1}{12} a^{11} + \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{360} a^{16} + \frac{1}{120} a^{15} + \frac{1}{72} a^{14} - \frac{7}{180} a^{13} - \frac{1}{180} a^{12} - \frac{17}{180} a^{11} + \frac{1}{18} a^{10} + \frac{1}{12} a^{9} + \frac{1}{9} a^{8} - \frac{1}{30} a^{7} + \frac{11}{45} a^{6} - \frac{1}{6} a^{5} + \frac{19}{90} a^{4} + \frac{22}{45} a^{3} - \frac{2}{15} a^{2} + \frac{11}{45} a - \frac{22}{45}$, $\frac{1}{360} a^{17} - \frac{1}{90} a^{15} + \frac{1}{360} a^{14} + \frac{1}{36} a^{13} + \frac{1}{180} a^{12} + \frac{1}{180} a^{11} + \frac{1}{12} a^{10} - \frac{5}{36} a^{9} - \frac{1}{5} a^{8} - \frac{7}{45} a^{7} - \frac{7}{30} a^{6} + \frac{19}{90} a^{5} - \frac{13}{90} a^{4} + \frac{1}{15} a^{3} - \frac{16}{45} a^{2} - \frac{2}{9} a - \frac{1}{5}$, $\frac{1}{4498920} a^{18} - \frac{1637}{1499640} a^{17} + \frac{1847}{4498920} a^{16} - \frac{21241}{2249460} a^{15} + \frac{21211}{1124730} a^{14} + \frac{11911}{562365} a^{13} - \frac{29053}{1124730} a^{12} + \frac{967}{62485} a^{11} - \frac{10199}{449892} a^{10} + \frac{40399}{374910} a^{9} - \frac{100468}{562365} a^{8} - \frac{1265}{37491} a^{7} - \frac{724}{562365} a^{6} - \frac{59456}{562365} a^{5} - \frac{31207}{187455} a^{4} - \frac{18022}{562365} a^{3} - \frac{28343}{112473} a^{2} - \frac{51061}{187455} a - \frac{42926}{187455}$, $\frac{1}{386907120} a^{19} - \frac{1}{64484520} a^{18} - \frac{2119}{19345356} a^{17} - \frac{184097}{193453560} a^{16} + \frac{121379}{96726780} a^{15} + \frac{143161}{10747420} a^{14} - \frac{3078361}{96726780} a^{13} - \frac{2146249}{96726780} a^{12} - \frac{1170926}{24181695} a^{11} + \frac{3868787}{96726780} a^{10} - \frac{2128571}{16121130} a^{9} - \frac{5381527}{48363390} a^{8} + \frac{701351}{16121130} a^{7} + \frac{1574941}{16121130} a^{6} - \frac{1413727}{9672678} a^{5} + \frac{645089}{16121130} a^{4} + \frac{249958}{537371} a^{3} - \frac{8701492}{24181695} a^{2} - \frac{601832}{1612113} a + \frac{1820701}{24181695}$

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:  $9$
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:  \( 8815064.44104 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.148877.1, 5.1.2382032.2 x5, 10.2.300726051798272.2 x5

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

Sibling fields

Degree 5 sibling: 5.1.2382032.2
Degree 10 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$