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Properties

Label 20.0.12664758105...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{31}\cdot 11^{10}$
Root discriminant $80.38$
Ramified primes $2, 5, 11$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2074993968080, 0, 235794769100, 0, -37512804175, 0, -5846151300, 0, 88578050, 0, 43483770, 0, 1976535, 0, 39930, 0, 1815, 0, 55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 55*x^18 + 1815*x^16 + 39930*x^14 + 1976535*x^12 + 43483770*x^10 + 88578050*x^8 - 5846151300*x^6 - 37512804175*x^4 + 235794769100*x^2 + 2074993968080)
 
gp: K = bnfinit(x^20 + 55*x^18 + 1815*x^16 + 39930*x^14 + 1976535*x^12 + 43483770*x^10 + 88578050*x^8 - 5846151300*x^6 - 37512804175*x^4 + 235794769100*x^2 + 2074993968080, 1)
 

Normalized defining polynomial

\( x^{20} + 55 x^{18} + 1815 x^{16} + 39930 x^{14} + 1976535 x^{12} + 43483770 x^{10} + 88578050 x^{8} - 5846151300 x^{6} - 37512804175 x^{4} + 235794769100 x^{2} + 2074993968080 \)

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:  \(126647581059570312500000000000000000000=2^{20}\cdot 5^{31}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.38$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $\frac{1}{11} a^{2}$, $\frac{1}{11} a^{3}$, $\frac{1}{121} a^{4}$, $\frac{1}{121} a^{5}$, $\frac{1}{1331} a^{6}$, $\frac{1}{1331} a^{7}$, $\frac{1}{14641} a^{8}$, $\frac{1}{14641} a^{9}$, $\frac{1}{161051} a^{10}$, $\frac{1}{161051} a^{11}$, $\frac{1}{1771561} a^{12}$, $\frac{1}{1771561} a^{13}$, $\frac{1}{19487171} a^{14}$, $\frac{1}{19487171} a^{15}$, $\frac{1}{643076643} a^{16} + \frac{1}{5314683} a^{12} + \frac{1}{483153} a^{10} - \frac{1}{43923} a^{8} + \frac{1}{363} a^{4} + \frac{1}{33} a^{2} + \frac{1}{3}$, $\frac{1}{1286153286} a^{17} - \frac{1}{38974342} a^{15} + \frac{1}{10629366} a^{13} - \frac{1}{483153} a^{11} - \frac{1}{87846} a^{9} - \frac{1}{363} a^{5} - \frac{1}{33} a^{3} + \frac{1}{6} a$, $\frac{1}{50504260859083448868} a^{18} + \frac{2036262541}{4591296441734858988} a^{16} + \frac{2475704179}{417390585612259908} a^{14} + \frac{3331366495}{18972299346011814} a^{12} + \frac{59179267}{104530574909156} a^{10} - \frac{2800690631}{156795862363734} a^{8} - \frac{1484233081}{14254169305794} a^{6} - \frac{1605116785}{647916786627} a^{4} + \frac{6185082881}{235606104228} a^{2} + \frac{717859552}{5354684187}$, $\frac{1}{101008521718166897736} a^{19} + \frac{2036262541}{9182592883469717976} a^{17} - \frac{18943032569}{834781171224519816} a^{15} + \frac{3331366495}{37944598692023628} a^{13} - \frac{6488606979}{2299672648001432} a^{11} + \frac{7908677743}{313591724727468} a^{9} + \frac{838648663}{2591667146508} a^{7} - \frac{1605116785}{1295833573254} a^{5} + \frac{6185082881}{471212208456} a^{3} - \frac{4636824635}{10709368374} a$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.242000.2, 5.1.78125.1, 10.2.30517578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
$11$11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$