# Properties

 Label 2496p1 Conductor $2496$ Discriminant $132844188672$ j-invariant $$\frac{1909913257984}{129730653}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3+x^2-2605x+47219$$ y^2=x^3+x^2-2605x+47219 (homogenize, simplify) $$y^2z=x^3+x^2z-2605xz^2+47219z^3$$ y^2z=x^3+x^2z-2605xz^2+47219z^3 (dehomogenize, simplify) $$y^2=x^3-211032x+35055720$$ y^2=x^3-211032x+35055720 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, -2605, 47219])

gp: E = ellinit([0, 1, 0, -2605, 47219])

magma: E := EllipticCurve([0, 1, 0, -2605, 47219]);

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);

 $P$ = $$\left(14, 117\right)$$ (14, 117) $\hat{h}(P)$ ≈ $0.20337571399893021702345126578$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(23, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-58,\pm 81)$$, $$(-25,\pm 312)$$, $$(14,\pm 117)$$, $$\left(23, 0\right)$$, $$(35,\pm 12)$$, $$(131,\pm 1404)$$

## 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: $132844188672$ = $2^{10} \cdot 3^{10} \cdot 13^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1909913257984}{129730653}$$ = $2^{11} \cdot 3^{-10} \cdot 13^{-3} \cdot 977^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.88295914252769068363078833141\dots$ Stable Faltings height: $0.30533649206106959244976156353\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.20337571399893021702345126578\dots$ 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: $1.0194039138474396958723049297\dots$ 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: $60$  = $2\cdot( 2 \cdot 5 )\cdot3$ 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: 3840 $\Gamma_0(N)$-optimal: yes 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$(j)_{-}$
$2$ $2$ $I_0^{*}$ Additive 1 6 10 0
$3$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$13$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

## 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], [53, 4, 106, 9], [134, 1, 11, 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], [53, 4, 106, 9], [134, 1, 11, 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

$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 53 & 4 \\ 106 & 9 \end{array}\right),\left(\begin{array}{rr} 134 & 1 \\ 11 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 153 & 4 \\ 152 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 41 & 118 \\ 116 & 39 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$

## $p$-adic regulators

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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

## 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 $K$ $2$ $$\Q(\sqrt{13})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.0.7488.3 $$\Z/4\Z$$ Not in database $8$ 8.0.9475854336.9 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.4.2846967791616.28 $$\Z/2\Z \oplus \Z/4\Z$$ 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.