# Properties

 Label 474320.br1 Conductor $474320$ Discriminant $7.778\times 10^{14}$ j-invariant $$\frac{2185454}{625}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -29080, 1347828])

gp: E = ellinit([0, 1, 0, -29080, 1347828])

magma: E := EllipticCurve([0, 1, 0, -29080, 1347828]);

## Simplified equation

 $$y^2=x^3+x^2-29080x+1347828$$ y^2=x^3+x^2-29080x+1347828 (homogenize, simplify) $$y^2z=x^3+x^2z-29080xz^2+1347828z^3$$ y^2z=x^3+x^2z-29080xz^2+1347828z^3 (dehomogenize, simplify) $$y^2=x^3-2355507x+989633106$$ y^2=x^3-2355507x+989633106 (homogenize, minimize)

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-4, 1210\right)$$ (-4, 1210) $$\left(-169, 1210\right)$$ (-169, 1210) $\hat{h}(P)$ ≈ $0.90071563153644120578487332205$ $1.9979690663069686903267678746$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(51, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-169,\pm 1210)$$, $$(-124,\pm 1750)$$, $$(-12,\pm 1302)$$, $$(-4,\pm 1210)$$, $$\left(51, 0\right)$$, $$(151,\pm 650)$$, $$(172,\pm 1210)$$, $$(436,\pm 8470)$$, $$(98611,\pm 30966320)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$474320$$ = $2^{4} \cdot 5 \cdot 7^{2} \cdot 11^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $777786141440000$ = $2^{11} \cdot 5^{4} \cdot 7^{3} \cdot 11^{6}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2185454}{625}$$ = $2 \cdot 5^{-4} \cdot 103^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.5638428833105486512076975289\dots$ Stable Faltings height: $-0.75696720586574814739874189061\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.6905885918118215260560467230\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.46921138317369852570968280429\dots$ 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: $64$  = $2\cdot2^{2}\cdot2\cdot2^{2}$ 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: $L^{(2)}(E,1)/2!$ ≈ $12.691894584667199999508005565$

## Modular invariants

Modular form 474320.2.a.br

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1966080 $\Gamma_0(N)$-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 474320.br2 is optimal.

## 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$(j)_{-}$
$2$ $2$ $I_{3}^{*}$ Additive 1 4 11 0
$5$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$7$ $2$ $III$ Additive -1 2 3 0
$11$ $4$ $I_0^{*}$ Additive -1 2 6 0

## 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 8.6.0.3

## $p$-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 474320.br consists of 2 curves linked by isogenies of degree 2.