Properties

 Label 2496p2 Conductor $2496$ Discriminant $-1.922\times 10^{13}$ j-invariant $$\frac{77366117936}{1172914587}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Simplified equation

 $$y^2=x^3+x^2+2255x+207599$$ y^2=x^3+x^2+2255x+207599 (homogenize, simplify) $$y^2z=x^3+x^2z+2255xz^2+207599z^3$$ y^2z=x^3+x^2z+2255xz^2+207599z^3 (dehomogenize, simplify) $$y^2=x^3+182628x+150791760$$ y^2=x^3+182628x+150791760 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, 2255, 207599])

gp: E = ellinit([0, 1, 0, 2255, 207599])

magma: E := EllipticCurve([0, 1, 0, 2255, 207599]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(5, 468\right)$$ (5, 468) $\hat{h}(P)$ ≈ $0.40675142799786043404690253157$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-47, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-47, 0\right)$$, $$(-46,\pm 93)$$, $$(5,\pm 468)$$, $$(122,\pm 1521)$$, $$(161,\pm 2184)$$, $$(8741,\pm 817284)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$2496$$ = $2^{6} \cdot 3 \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-19217032593408$ = $-1 \cdot 2^{14} \cdot 3^{5} \cdot 13^{6}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{77366117936}{1172914587}$$ = $2^{4} \cdot 3^{-5} \cdot 13^{-6} \cdot 19^{3} \cdot 89^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.2295327328076633383394043921\dots$ Stable Faltings height: $0.42086102215439381068596691707\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.40675142799786043404690253157\dots$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $0.50970195692371984793615246483\dots$ 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: $60$  = $2\cdot5\cdot( 2 \cdot 3 )$ 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: $1$ (exact) 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); Special value: $L'(E,1)$ ≈ $3.1098299824804049140006758666$

Modular invariants

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);

$$q + q^{3} - 4 q^{5} + q^{9} + 2 q^{11} + q^{13} - 4 q^{15} + 2 q^{17} - 8 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 7680 $\Gamma_0(N)$-optimal: no Manin constant: 1

Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$2$ $2$ $I_{4}^{*}$ Additive 1 6 14 0
$3$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$13$ $6$ $I_{6}$ Split multiplicative -1 1 6 6

Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 2.3.0.1
sage: gens = [[1, 2, 2, 5], [145, 4, 134, 9], [106, 1, 103, 0], [3, 4, 8, 11], [153, 4, 152, 5], [1, 4, 0, 1], [41, 118, 116, 39], [1, 0, 4, 1]]

sage: GL(2,Integers(156)).subgroup(gens)

magma: Gens := [[1, 2, 2, 5], [145, 4, 134, 9], [106, 1, 103, 0], [3, 4, 8, 11], [153, 4, 152, 5], [1, 4, 0, 1], [41, 118, 116, 39], [1, 0, 4, 1]];

magma: sub<GL(2,Integers(156))|Gens>;

The image of the adelic Galois representation has level $156$, index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 145 & 4 \\ 134 & 9 \end{array}\right),\left(\begin{array}{rr} 106 & 1 \\ 103 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 153 & 4 \\ 152 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 41 & 118 \\ 116 & 39 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$

$p$-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.

Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add split ord ss ord split ord ord ord ord ord ord ord ord ord - 2 7 1,1 1 2 1 1 1 1 1 1 1 1 1 - 0 0 0,0 0 0 0 0 0 0 0 0 0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 2496p 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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.2.32448.1 $$\Z/4\Z$$ Not in database $8$ 8.0.8074100736.36 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.9475854336.8 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.2.11609505792.5 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.