Properties

Label 296208.d1
Conductor $296208$
Discriminant $3.229\times 10^{18}$
j-invariant \( \frac{832972004929}{610368} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -3415467, 2427996890])
 
gp: E = ellinit([0, 0, 0, -3415467, 2427996890])
 
magma: E := EllipticCurve([0, 0, 0, -3415467, 2427996890]);
 

\(y^2=x^3-3415467x+2427996890\)  Toggle raw display

Mordell-Weil group structure

\(\Z^2 \times \Z/{2}\Z\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \(\left(1111, 2178\right)\)  Toggle raw display\(\left(319, 37026\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $0.99179897069911471733529597600$$1.2566963832799884370031457619$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(1045, 0\right) \)  Toggle raw display

Integral points

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

\((-2129,\pm 7038)\), \((-1067,\pm 69696)\), \((319,\pm 37026)\), \((506,\pm 28798)\), \((917,\pm 8192)\), \( \left(1045, 0\right) \), \((1111,\pm 2178)\), \((1126,\pm 3132)\), \((1237,\pm 9792)\), \((2134,\pm 69696)\), \((6677,\pm 526592)\), \((123079,\pm 43174494)\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 296208 \)  =  \(2^{4} \cdot 3^{2} \cdot 11^{2} \cdot 17\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(3228756874455416832 \)  =  \(2^{18} \cdot 3^{7} \cdot 11^{7} \cdot 17^{2} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{832972004929}{610368} \)  =  \(2^{-6} \cdot 3^{-1} \cdot 11^{-1} \cdot 17^{-2} \cdot 97^{6}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(2.4857688253610796182265377017\dots\)
Stable Faltings height: \(0.044367864067894191080711172798\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1.2210396742409595243070848070\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.24968502087225768408890125633\dots\)
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: \( 128 \)  = \( 2^{2}\cdot2^{2}\cdot2^{2}\cdot2 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(2\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 296208.2.a.d

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);
 

\( q - 4q^{5} - 2q^{7} - q^{17} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 8847360
\( \Gamma_0(N) \)-optimal: yes
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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 9.7560101295586784665565033551402832295 \)

Local data

This elliptic curve is not semistable. There are 4 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\((j)_{-}\)
\(2\) \(4\) \(I_{10}^{*}\) Additive -1 4 18 6
\(3\) \(4\) \(I_1^{*}\) Additive -1 2 7 1
\(11\) \(4\) \(I_1^{*}\) Additive -1 2 7 1
\(17\) \(2\) \(I_{2}\) Non-split multiplicative 1 1 2 2

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X15.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 2 & 1 \end{array}\right)$ and has index 6.

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
\(2\) B

$p$-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 \(d=\) 2.
Its isogeny class 296208.d consists of 2 curves linked by isogenies of degree 2.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{33}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$4$ 4.0.8448.2 \(\Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.77720518656.13 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) 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.