Properties

Label 203280.ds1
Conductor $203280$
Discriminant $-3.183\times 10^{27}$
j-invariant \( -\frac{2126464142970105856}{438611057788643355} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -51866085, -2718075836595])
 
gp: E = ellinit([0, -1, 0, -51866085, -2718075836595])
 
magma: E := EllipticCurve([0, -1, 0, -51866085, -2718075836595]);
 

\(y^2=x^3-x^2-51866085x-2718075836595\)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

\(P\) =  \( \left(39692, 7599647\right) \)
\(\hat{h}(P)\) ≈  $8.4052688058246229872315506891$

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\((39692,\pm 7599647)\)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 203280 \)  =  \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-3182699496026549496328826880 \)  =  \(-1 \cdot 2^{12} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{21} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{2126464142970105856}{438611057788643355} \)  =  \(-1 \cdot 2^{12} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{-15} \cdot 179^{3} \cdot 449^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(8.4052688058246229872315506891\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.020000894867476545796315619116\)
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: \( 2 \)  = \( 1\cdot1\cdot1\cdot1\cdot2 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(1\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(25\) = $5^2$ (exact)

Modular invariants

Modular form 203280.2.a.ds

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

\( q - q^{3} + q^{5} + q^{7} + q^{9} + 6q^{13} - q^{15} + 7q^{17} - 5q^{19} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 144000000
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 

\( L'(E,1) \) ≈ \( 8.4056448859089208561924128312077918293 \)

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(1\) \(II^{*}\) Additive -1 4 12 0
\(3\) \(1\) \(I_{1}\) Non-split multiplicative 1 1 1 1
\(5\) \(1\) \(I_{1}\) Split multiplicative -1 1 1 1
\(7\) \(1\) \(I_{1}\) Split multiplicative -1 1 1 1
\(11\) \(2\) \(I_{15}^{*}\) Additive -1 2 21 15

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(5\) B.4.2

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 203280.ds consists of 2 curves linked by isogenies of degree 5.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.4620.1 \(\Z/2\Z\) Not in database
$4$ 4.0.242000.2 \(\Z/5\Z\) Not in database
$6$ 6.0.24652782000.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$8$ Deg 8 \(\Z/3\Z\) Not in database
$10$ 10.2.23794567667767230468750000000000.1 \(\Z/5\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$12$ Deg 12 \(\Z/10\Z\) Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.