# Properties

 Label 485100.d1 Conductor $485100$ Discriminant $4.161\times 10^{19}$ j-invariant $$\frac{59466754384}{121275}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -5692575, 5218487750])

gp: E = ellinit([0, 0, 0, -5692575, 5218487750])

magma: E := EllipticCurve([0, 0, 0, -5692575, 5218487750]);

$$y^2=x^3-5692575x+5218487750$$

## Mordell-Weil group structure

$\Z^2 \times \Z/{2}\Z$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(2555, 85750\right)$$ $$\left(1555, 11250\right)$$ $\hat{h}(P)$ ≈ $1.3999292876383519842855012648$ $2.3207817956340364714392731015$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(1330, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-2170,\pm 85750)$$, $$(-385,\pm 85750)$$, $$(1015,\pm 22050)$$, $$\left(1330, 0\right)$$, $$(1451,\pm 3674)$$, $$(1555,\pm 11250)$$, $$(2555,\pm 85750)$$, $$(5299,\pm 351918)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$485100$$ = $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $41605145297100000000$ = $2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{59466754384}{121275}$$ = $2^{4} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 11^{-1} \cdot 1549^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.6510499488776880838701234333\dots$ Stable Faltings height: $-0.13802834657437047462537663780\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $3.2439720621187625200319542825\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.20382698889648965045041067002\dots$ 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: $96$  = $3\cdot2\cdot2^{2}\cdot2^{2}\cdot1$ 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) 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: $L^{(2)}(E,1)/2!$ ≈ $15.869017379664087426843604925006347925$

## Modular invariants

Modular form 485100.2.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);

$$q - q^{11} - 6q^{13} - 2q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 17694720 $\Gamma_0(N)$-optimal: not computed* (one of 2 curves in this isogeny class which might be optimal) Manin constant: 1 (conditional*)
* The optimal curve in each isogeny class has not been determined in all cases for conductors over 400000. The Manin constant is correct provided that curve 485100.d2 is optimal.

## 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$(j)_{-}$
$2$ $3$ $IV^{*}$ Additive -1 2 8 0
$3$ $2$ $I_2^{*}$ Additive -1 2 8 2
$5$ $4$ $I_2^{*}$ Additive 1 2 8 2
$7$ $4$ $I_2^{*}$ Additive -1 2 8 2
$11$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## 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 $\GL(2,\Z_\ell)$ 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

## $p$-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 485100.d consists of 2 curves linked by isogenies of degree 2.