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Properties

Label 20.4.48414218021...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{18}\cdot 5^{12}\cdot 13^{15}$
Root discriminant $48.33$
Ramified primes $3, 5, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-274401, 1160244, -777600, 371223, -285120, 183780, -9099, -42075, 25146, -28629, 18903, -1209, -2445, 246, 135, -36, 9, 27, -9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 9*x^18 + 27*x^17 + 9*x^16 - 36*x^15 + 135*x^14 + 246*x^13 - 2445*x^12 - 1209*x^11 + 18903*x^10 - 28629*x^9 + 25146*x^8 - 42075*x^7 - 9099*x^6 + 183780*x^5 - 285120*x^4 + 371223*x^3 - 777600*x^2 + 1160244*x - 274401)
 
gp: K = bnfinit(x^20 - 3*x^19 - 9*x^18 + 27*x^17 + 9*x^16 - 36*x^15 + 135*x^14 + 246*x^13 - 2445*x^12 - 1209*x^11 + 18903*x^10 - 28629*x^9 + 25146*x^8 - 42075*x^7 - 9099*x^6 + 183780*x^5 - 285120*x^4 + 371223*x^3 - 777600*x^2 + 1160244*x - 274401, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 9 x^{18} + 27 x^{17} + 9 x^{16} - 36 x^{15} + 135 x^{14} + 246 x^{13} - 2445 x^{12} - 1209 x^{11} + 18903 x^{10} - 28629 x^{9} + 25146 x^{8} - 42075 x^{7} - 9099 x^{6} + 183780 x^{5} - 285120 x^{4} + 371223 x^{3} - 777600 x^{2} + 1160244 x - 274401 \)

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:  \(4841421802104669181474651611328125=3^{18}\cdot 5^{12}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.33$
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:  $3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{18} a^{15} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2}$, $\frac{1}{18} a^{16} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a$, $\frac{1}{18} a^{17} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{36} a^{18} - \frac{1}{36} a^{17} - \frac{1}{12} a^{14} - \frac{1}{12} a^{10} - \frac{5}{12} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8607558858597236830021194103721140849464041221752} a^{19} + \frac{39643778148941392247163754646386556941887749221}{4303779429298618415010597051860570424732020610876} a^{18} + \frac{19804795598305856853531343116559717892440629959}{2869186286199078943340398034573713616488013740584} a^{17} + \frac{13167852102234654711384365128485857779563203047}{1075944857324654603752649262965142606183005152719} a^{16} + \frac{44166604173874019937280538424543687605424141251}{2869186286199078943340398034573713616488013740584} a^{15} + \frac{1027790003452786462206103198974779627312726323}{168775663894063467255317531445512565675765514152} a^{14} - \frac{53895720311452285017440797207422094355821590655}{717296571549769735835099508643428404122003435146} a^{13} - \frac{24750037309563614410296071465532948461236009283}{478197714366513157223399672428952269414668956764} a^{12} - \frac{220199486111545493504371333346778904896121215905}{2869186286199078943340398034573713616488013740584} a^{11} + \frac{12206422521057078520449761334783639607997557965}{119549428591628289305849918107238067353667239191} a^{10} + \frac{475790582441501581014731122864023597143833120477}{2869186286199078943340398034573713616488013740584} a^{9} + \frac{70392876374667996530017286392404139772214909953}{1434593143099539471670199017286856808244006870292} a^{8} + \frac{140391662632232814948535529975483410267220301650}{358648285774884867917549754321714202061001717573} a^{7} + \frac{862303499040402894807436043682116977214562316103}{2869186286199078943340398034573713616488013740584} a^{6} + \frac{9064444615558214315987338345526613907639545415}{478197714366513157223399672428952269414668956764} a^{5} + \frac{225023272604223750514146082415054698534259225265}{478197714366513157223399672428952269414668956764} a^{4} - \frac{6435701590865350760421243010627635984359808687}{28129277315677244542552921907585427612627585692} a^{3} - \frac{327920104005041342241472802712838176278583983939}{956395428733026314446799344857904538829337913528} a^{2} - \frac{1411356089078413237984280970839942379935982099}{956395428733026314446799344857904538829337913528} a - \frac{115656820019865516156550231735310321647482974883}{956395428733026314446799344857904538829337913528}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1022414654.5228317 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.19773.1, 5.1.1711125.1, 10.2.38063333953125.2

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ $20$

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
$3$3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
5Data not computed
13Data not computed