↑

Properties

Label 20.0.51888244744...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{8}\cdot 5^{31}\cdot 19^{8}$
Root discriminant $61.06$
Ramified primes $3, 5, 19$
Class number $109$ (GRH)
Class group $[109]$ (GRH)
Galois group $C_2\times C_5:F_5$ (as 20T49)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1805, 0, 27075, 0, 162450, 0, 495425, 0, 804650, 0, 666510, 0, 252865, 0, 42350, 0, 2955, 0, 90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 90*x^18 + 2955*x^16 + 42350*x^14 + 252865*x^12 + 666510*x^10 + 804650*x^8 + 495425*x^6 + 162450*x^4 + 27075*x^2 + 1805)
 
gp: K = bnfinit(x^20 + 90*x^18 + 2955*x^16 + 42350*x^14 + 252865*x^12 + 666510*x^10 + 804650*x^8 + 495425*x^6 + 162450*x^4 + 27075*x^2 + 1805, 1)
 

Normalized defining polynomial

\( x^{20} + 90 x^{18} + 2955 x^{16} + 42350 x^{14} + 252865 x^{12} + 666510 x^{10} + 804650 x^{8} + 495425 x^{6} + 162450 x^{4} + 27075 x^{2} + 1805 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(518882447443907149136066436767578125=3^{8}\cdot 5^{31}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.06$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{10} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{114} a^{15} - \frac{5}{114} a^{13} - \frac{3}{38} a^{11} - \frac{10}{57} a^{9} - \frac{1}{2} a^{8} - \frac{22}{57} a^{7} - \frac{1}{2} a^{6} - \frac{29}{114} a^{5} - \frac{1}{3} a$, $\frac{1}{114} a^{16} - \frac{5}{114} a^{14} - \frac{3}{38} a^{12} - \frac{10}{57} a^{10} + \frac{13}{114} a^{8} - \frac{1}{2} a^{7} - \frac{29}{114} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{114} a^{17} + \frac{2}{57} a^{13} + \frac{11}{114} a^{11} - \frac{11}{114} a^{9} - \frac{7}{38} a^{7} - \frac{25}{57} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{661740720046428} a^{18} - \frac{2630799979319}{661740720046428} a^{16} + \frac{230002183072}{15039561819237} a^{14} + \frac{1708227490709}{30079123638474} a^{12} + \frac{4355089683095}{661740720046428} a^{10} + \frac{19659177033773}{60158247276948} a^{8} - \frac{1}{2} a^{7} + \frac{1577510299981}{3166223540892} a^{6} - \frac{1}{2} a^{5} - \frac{1084537836862}{8707114737453} a^{4} + \frac{15080651200}{8707114737453} a^{2} - \frac{1}{2} a + \frac{2792443639423}{11609486316604}$, $\frac{1}{1323481440092856} a^{19} - \frac{1}{1323481440092856} a^{18} + \frac{1057981059661}{441160480030952} a^{17} - \frac{1057981059661}{441160480030952} a^{16} + \frac{65384134801}{20052749092316} a^{15} + \frac{859255442561}{60158247276948} a^{14} + \frac{2041447573189}{30079123638474} a^{13} + \frac{166610041240}{15039561819237} a^{12} - \frac{70140135724693}{441160480030952} a^{11} - \frac{219130586540269}{1323481440092856} a^{10} - \frac{5670611293363}{120316494553896} a^{9} + \frac{1186598533145}{40105498184632} a^{8} - \frac{16042494991899}{40105498184632} a^{7} + \frac{15409841719813}{120316494553896} a^{6} - \frac{102162240962927}{661740720046428} a^{5} - \frac{6538039063729}{34828458949812} a^{4} + \frac{2917452230351}{17414229474906} a^{3} - \frac{969970634917}{2902371579151} a^{2} - \frac{14841641714939}{69656917899624} a + \frac{8817042677181}{23218972633208}$

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

Class group and class number

$C_{109}$, which has order $109$ (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:  \( -\frac{1041138233112565}{1323481440092856} a^{19} - \frac{2608216055}{8051892024} a^{18} - \frac{1638033039162219}{23218972633208} a^{17} - \frac{233903129045}{8051892024} a^{16} - \frac{46160007388753783}{20052749092316} a^{15} - \frac{346921690595}{365995092} a^{14} - \frac{25785406886248370}{791555885223} a^{13} - \frac{1227393157625}{91498773} a^{12} - \frac{83137539971595372995}{441160480030952} a^{11} - \frac{624890478886087}{8051892024} a^{10} - \frac{55797364244592811265}{120316494553896} a^{9} - \frac{46605866807605}{243996728} a^{8} - \frac{19407737733038135165}{40105498184632} a^{7} - \frac{145939083525805}{731990184} a^{6} - \frac{154923086482495147543}{661740720046428} a^{5} - \frac{388471206696475}{4025946012} a^{4} - \frac{913538441963261285}{17414229474906} a^{3} - \frac{43542337909085}{2012973006} a^{2} - \frac{306878050117048105}{69656917899624} a - \frac{14630705194129}{8051892024} \) (order $10$)
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:  \( 307931905.969 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_5:F_5$ (as 20T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 20 conjugacy class representatives for $C_2\times C_5:F_5$
Character table for $C_2\times C_5:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.10.322143585205078125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$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$$[\ ]_{2}^{4}$
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.10.8.1$x^{10} - 209 x^{5} + 11552$$5$$2$$8$$D_5$$[\ ]_{5}^{2}$