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Properties

Label 20.0.86825139158...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{15}\cdot 11^{16}$
Root discriminant $78.87$
Ramified primes $2, 3, 5, 11$
Class number $500$ (GRH)
Class group $[5, 10, 10]$ (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![106583445, 0, -45074070, 0, 9191961, 0, 896670, 0, -629856, 0, 572265, 0, -122229, 0, 16740, 0, -1161, 0, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 15*x^18 - 1161*x^16 + 16740*x^14 - 122229*x^12 + 572265*x^10 - 629856*x^8 + 896670*x^6 + 9191961*x^4 - 45074070*x^2 + 106583445)
 
gp: K = bnfinit(x^20 + 15*x^18 - 1161*x^16 + 16740*x^14 - 122229*x^12 + 572265*x^10 - 629856*x^8 + 896670*x^6 + 9191961*x^4 - 45074070*x^2 + 106583445, 1)
 

Normalized defining polynomial

\( x^{20} + 15 x^{18} - 1161 x^{16} + 16740 x^{14} - 122229 x^{12} + 572265 x^{10} - 629856 x^{8} + 896670 x^{6} + 9191961 x^{4} - 45074070 x^{2} + 106583445 \)

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:  \(86825139158850321116448000000000000000=2^{20}\cdot 3^{10}\cdot 5^{15}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.87$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{2673} a^{10} - \frac{1}{297} a^{8} - \frac{1}{99} a^{6} + \frac{4}{99} a^{4} + \frac{1}{33} a^{2} - \frac{1}{11}$, $\frac{1}{2673} a^{11} - \frac{1}{297} a^{9} - \frac{1}{99} a^{7} + \frac{4}{99} a^{5} + \frac{1}{33} a^{3} - \frac{1}{11} a$, $\frac{1}{24057} a^{12} + \frac{10}{2673} a^{8} - \frac{16}{891} a^{6} + \frac{2}{297} a^{4} - \frac{1}{11} a^{2} + \frac{8}{33}$, $\frac{1}{24057} a^{13} + \frac{10}{2673} a^{9} - \frac{16}{891} a^{7} + \frac{2}{297} a^{5} - \frac{1}{11} a^{3} + \frac{8}{33} a$, $\frac{1}{7289271} a^{14} + \frac{2}{809919} a^{12} + \frac{5}{73629} a^{10} - \frac{58}{269973} a^{8} - \frac{361}{89991} a^{6} - \frac{445}{9999} a^{4} - \frac{1288}{9999} a^{2} + \frac{430}{1111}$, $\frac{1}{7289271} a^{15} + \frac{2}{809919} a^{13} + \frac{5}{73629} a^{11} - \frac{58}{269973} a^{9} - \frac{361}{89991} a^{7} - \frac{445}{9999} a^{5} - \frac{1288}{9999} a^{3} + \frac{430}{1111} a$, $\frac{1}{65603439} a^{16} + \frac{1}{21867813} a^{14} + \frac{14}{809919} a^{12} - \frac{10}{809919} a^{10} + \frac{1121}{269973} a^{8} - \frac{685}{89991} a^{6} + \frac{5}{3333} a^{4} + \frac{3791}{29997} a^{2} - \frac{2612}{9999}$, $\frac{1}{1246465341} a^{17} + \frac{25}{415488447} a^{15} - \frac{172}{15388461} a^{13} - \frac{1994}{15388461} a^{11} - \frac{9376}{1709829} a^{9} - \frac{1090}{569943} a^{7} - \frac{2764}{63327} a^{5} + \frac{11966}{569943} a^{3} + \frac{2825}{17271} a$, $\frac{1}{57029528746773} a^{18} - \frac{2855}{576055845927} a^{16} - \frac{136144}{6336614305197} a^{14} + \frac{5020732}{704068256133} a^{12} - \frac{23152952}{234689418711} a^{10} + \frac{283961686}{78229806237} a^{8} - \frac{466700387}{26076602079} a^{6} + \frac{1330324079}{26076602079} a^{4} + \frac{1368823793}{8692200693} a^{2} - \frac{55376971}{152494749}$, $\frac{1}{57029528746773} a^{19} - \frac{301}{2112204768399} a^{17} + \frac{412892}{6336614305197} a^{15} + \frac{4286582}{234689418711} a^{13} + \frac{2878835}{26076602079} a^{11} + \frac{436227670}{78229806237} a^{9} - \frac{117281702}{8692200693} a^{7} + \frac{334464281}{26076602079} a^{5} + \frac{1015519127}{8692200693} a^{3} + \frac{163174490}{2897400231} a$

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

Class group and class number

$C_{5}\times C_{10}\times C_{10}$, which has order $500$ (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:  \( 212425982.34586897 \) (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

\(\Q(\sqrt{5}) \), 4.0.18000.1, 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$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$$[2]^{4}$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$