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Properties

Label 20.4.37791360000...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{28}\cdot 3^{10}\cdot 5^{22}$
Root discriminant $26.85$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-239, 1080, -1260, 280, 225, -928, 2300, 120, -590, 800, -1432, -80, 70, -520, 0, -8, -55, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 55*x^16 - 8*x^15 - 520*x^13 + 70*x^12 - 80*x^11 - 1432*x^10 + 800*x^9 - 590*x^8 + 120*x^7 + 2300*x^6 - 928*x^5 + 225*x^4 + 280*x^3 - 1260*x^2 + 1080*x - 239)
 
gp: K = bnfinit(x^20 - 55*x^16 - 8*x^15 - 520*x^13 + 70*x^12 - 80*x^11 - 1432*x^10 + 800*x^9 - 590*x^8 + 120*x^7 + 2300*x^6 - 928*x^5 + 225*x^4 + 280*x^3 - 1260*x^2 + 1080*x - 239, 1)
 

Normalized defining polynomial

\( x^{20} - 55 x^{16} - 8 x^{15} - 520 x^{13} + 70 x^{12} - 80 x^{11} - 1432 x^{10} + 800 x^{9} - 590 x^{8} + 120 x^{7} + 2300 x^{6} - 928 x^{5} + 225 x^{4} + 280 x^{3} - 1260 x^{2} + 1080 x - 239 \)

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:  \(37791360000000000000000000000=2^{28}\cdot 3^{10}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.85$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{9644} a^{18} + \frac{21}{4822} a^{17} + \frac{254}{2411} a^{16} - \frac{799}{9644} a^{15} - \frac{181}{4822} a^{14} + \frac{1389}{9644} a^{13} + \frac{609}{4822} a^{12} - \frac{2237}{9644} a^{11} - \frac{493}{2411} a^{10} + \frac{1523}{9644} a^{9} - \frac{157}{2411} a^{8} - \frac{267}{9644} a^{7} + \frac{1168}{2411} a^{6} + \frac{3067}{9644} a^{5} - \frac{652}{2411} a^{4} - \frac{813}{9644} a^{3} + \frac{2529}{9644} a^{2} - \frac{1445}{9644} a + \frac{2045}{4822}$, $\frac{1}{6467569216835351933256772764172} a^{19} - \frac{3117205612613834701400529}{1616892304208837983314193191043} a^{18} + \frac{560409215468476192579556471237}{6467569216835351933256772764172} a^{17} - \frac{57844666583281987229388958111}{6467569216835351933256772764172} a^{16} - \frac{291917041179486562067932446081}{6467569216835351933256772764172} a^{15} - \frac{72123249099378194686132837223}{6467569216835351933256772764172} a^{14} + \frac{1466191741394884552945424221757}{6467569216835351933256772764172} a^{13} - \frac{664929486074688173797209861673}{6467569216835351933256772764172} a^{12} - \frac{199018733248962973719729177153}{6467569216835351933256772764172} a^{11} - \frac{716278341438255078482696513401}{6467569216835351933256772764172} a^{10} + \frac{935498651240958730384041124453}{6467569216835351933256772764172} a^{9} + \frac{2765832649846884997759386227343}{6467569216835351933256772764172} a^{8} + \frac{2591883711943327088903637433667}{6467569216835351933256772764172} a^{7} - \frac{831795233662318422307333790617}{6467569216835351933256772764172} a^{6} + \frac{2471700131823413726021102418931}{6467569216835351933256772764172} a^{5} + \frac{1063121307674193074789614385339}{6467569216835351933256772764172} a^{4} - \frac{805606225464049592361554618419}{3233784608417675966628386382086} a^{3} - \frac{2556074824797272089152628477879}{6467569216835351933256772764172} a^{2} + \frac{1562235964883162543507821722803}{3233784608417675966628386382086} a + \frac{715734878765550177697518190161}{3233784608417675966628386382086}$

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:  $11$
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:  \( 2354906.8163907384 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

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

\(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}, \sqrt{5})\), 5.1.50000.1, 10.2.194400000000000.1, 10.2.38880000000000.1, 10.2.12500000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$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}$
5Data not computed