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Properties

Label 20.4.70536476880...0561.2
Degree $20$
Signature $[4, 8]$
Discriminant $3^{8}\cdot 401^{10}$
Root discriminant $31.08$
Ramified primes $3, 401$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^4:D_5$ (as 20T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51, 19, -186, -147, -686, 3137, -2813, 2361, -4986, 4072, -1083, 1536, -2250, 929, 337, -317, -6, 49, -4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 4*x^18 + 49*x^17 - 6*x^16 - 317*x^15 + 337*x^14 + 929*x^13 - 2250*x^12 + 1536*x^11 - 1083*x^10 + 4072*x^9 - 4986*x^8 + 2361*x^7 - 2813*x^6 + 3137*x^5 - 686*x^4 - 147*x^3 - 186*x^2 + 19*x + 51)
 
gp: K = bnfinit(x^20 - 5*x^19 - 4*x^18 + 49*x^17 - 6*x^16 - 317*x^15 + 337*x^14 + 929*x^13 - 2250*x^12 + 1536*x^11 - 1083*x^10 + 4072*x^9 - 4986*x^8 + 2361*x^7 - 2813*x^6 + 3137*x^5 - 686*x^4 - 147*x^3 - 186*x^2 + 19*x + 51, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 4 x^{18} + 49 x^{17} - 6 x^{16} - 317 x^{15} + 337 x^{14} + 929 x^{13} - 2250 x^{12} + 1536 x^{11} - 1083 x^{10} + 4072 x^{9} - 4986 x^{8} + 2361 x^{7} - 2813 x^{6} + 3137 x^{5} - 686 x^{4} - 147 x^{3} - 186 x^{2} + 19 x + 51 \)

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:  \(705364768808763611651745450561=3^{8}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.08$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{412594363787936452695584961359320599} a^{19} - \frac{205455163157862440128356088711194706}{412594363787936452695584961359320599} a^{18} - \frac{50522271429812785244384136064768289}{412594363787936452695584961359320599} a^{17} + \frac{72200472058670146881879797092317059}{412594363787936452695584961359320599} a^{16} - \frac{202654723023156822919396850786448007}{412594363787936452695584961359320599} a^{15} + \frac{21060006127016466701849954440224935}{137531454595978817565194987119773533} a^{14} - \frac{131126237406994722222510307200306674}{412594363787936452695584961359320599} a^{13} + \frac{60777139962040402656977747361062750}{137531454595978817565194987119773533} a^{12} - \frac{1493900259230817715338548832625932}{10579342661229139812707306701521041} a^{11} + \frac{42312772541723746584724981565094404}{137531454595978817565194987119773533} a^{10} + \frac{20779841544125812173639026853718894}{137531454595978817565194987119773533} a^{9} + \frac{33928774664087120180877095454800794}{412594363787936452695584961359320599} a^{8} + \frac{204302601218746973505907710071573251}{412594363787936452695584961359320599} a^{7} - \frac{31581072101804761367048025650918762}{412594363787936452695584961359320599} a^{6} + \frac{28922086982347872059956767006796754}{412594363787936452695584961359320599} a^{5} + \frac{9567185209846738164609014579834848}{412594363787936452695584961359320599} a^{4} + \frac{76460087296012147149453827685958145}{412594363787936452695584961359320599} a^{3} + \frac{31728187748156656543620776070989789}{412594363787936452695584961359320599} a^{2} + \frac{169388808141978442523938594406495993}{412594363787936452695584961359320599} a + \frac{53969605061737871967349121849446405}{137531454595978817565194987119773533}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 13089361.5818 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.6.839859969762081.2, 10.6.93317774418009.2, 10.2.232712654409.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$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}$
401Data not computed