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Properties

Label 20.4.14703913556...0000.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 5^{15}\cdot 11^{16}$
Root discriminant $45.54$
Ramified primes $2, 5, 11$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
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![1805, 0, 2290, 0, 1401, 0, -410, 0, -864, 0, -2355, 0, -1509, 0, -620, 0, -129, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 129*x^16 - 620*x^14 - 1509*x^12 - 2355*x^10 - 864*x^8 - 410*x^6 + 1401*x^4 + 2290*x^2 + 1805)
 
gp: K = bnfinit(x^20 - 5*x^18 - 129*x^16 - 620*x^14 - 1509*x^12 - 2355*x^10 - 864*x^8 - 410*x^6 + 1401*x^4 + 2290*x^2 + 1805, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} - 129 x^{16} - 620 x^{14} - 1509 x^{12} - 2355 x^{10} - 864 x^{8} - 410 x^{6} + 1401 x^{4} + 2290 x^{2} + 1805 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1470391355634309152000000000000000=2^{20}\cdot 5^{15}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.54$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} + \frac{3}{11} a^{8} - \frac{3}{11} a^{6} - \frac{4}{11} a^{4} + \frac{1}{11} a^{2} + \frac{1}{11}$, $\frac{1}{11} a^{11} + \frac{3}{11} a^{9} - \frac{3}{11} a^{7} - \frac{4}{11} a^{5} + \frac{1}{11} a^{3} + \frac{1}{11} a$, $\frac{1}{33} a^{12} + \frac{10}{33} a^{8} + \frac{16}{33} a^{6} + \frac{2}{33} a^{4} + \frac{3}{11} a^{2} + \frac{8}{33}$, $\frac{1}{33} a^{13} + \frac{10}{33} a^{9} + \frac{16}{33} a^{7} + \frac{2}{33} a^{5} + \frac{3}{11} a^{3} + \frac{8}{33} a$, $\frac{1}{3333} a^{14} - \frac{2}{1111} a^{12} + \frac{5}{303} a^{10} + \frac{58}{3333} a^{8} - \frac{361}{3333} a^{6} + \frac{445}{1111} a^{4} - \frac{1288}{3333} a^{2} - \frac{430}{1111}$, $\frac{1}{3333} a^{15} - \frac{2}{1111} a^{13} + \frac{5}{303} a^{11} + \frac{58}{3333} a^{9} - \frac{361}{3333} a^{7} + \frac{445}{1111} a^{5} - \frac{1288}{3333} a^{3} - \frac{430}{1111} a$, $\frac{1}{9999} a^{16} - \frac{1}{9999} a^{14} + \frac{14}{1111} a^{12} + \frac{10}{3333} a^{10} + \frac{1121}{3333} a^{8} + \frac{685}{3333} a^{6} + \frac{15}{1111} a^{4} - \frac{3791}{9999} a^{2} - \frac{2612}{9999}$, $\frac{1}{189981} a^{17} - \frac{25}{189981} a^{15} - \frac{172}{21109} a^{13} + \frac{1994}{63327} a^{11} - \frac{9376}{21109} a^{9} + \frac{1090}{21109} a^{7} - \frac{8292}{21109} a^{5} - \frac{11966}{189981} a^{3} + \frac{2825}{17271} a$, $\frac{1}{2897400231} a^{18} + \frac{2855}{87800007} a^{16} - \frac{136144}{2897400231} a^{14} - \frac{5020732}{965800077} a^{12} - \frac{23152952}{965800077} a^{10} - \frac{283961686}{965800077} a^{8} - \frac{466700387}{965800077} a^{6} - \frac{1330324079}{2897400231} a^{4} + \frac{1368823793}{2897400231} a^{2} + \frac{55376971}{152494749}$, $\frac{1}{2897400231} a^{19} + \frac{301}{321933359} a^{17} + \frac{412892}{2897400231} a^{15} - \frac{4286582}{321933359} a^{13} + \frac{8636505}{321933359} a^{11} - \frac{436227670}{965800077} a^{9} - \frac{117281702}{321933359} a^{7} - \frac{334464281}{2897400231} a^{5} + \frac{1015519127}{2897400231} a^{3} - \frac{163174490}{2897400231} a$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $11$
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:  \( 354914655.78466165 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 5.1.1830125.1, 10.2.16746787578125.1

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 ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$$[2]^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$