# Properties

 Label 784.b1 Conductor $784$ Discriminant $1.209\times 10^{13}$ j-invariant $$\frac{2251439055699625}{25088}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -2140728, -1206278060])

gp: E = ellinit([0, 1, 0, -2140728, -1206278060])

magma: E := EllipticCurve([0, 1, 0, -2140728, -1206278060]);

$$y^2=x^3+x^2-2140728x-1206278060$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(\frac{33604}{9}, \frac{5590214}{27}\right)$$ $\hat{h}(P)$ ≈ $7.0144143340292381702111744635$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-845, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-845, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$784$$ = $2^{4} \cdot 7^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $12089663946752$ = $2^{21} \cdot 7^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2251439055699625}{25088}$$ = $2^{-9} \cdot 5^{3} \cdot 7^{-2} \cdot 11^{3} \cdot 2383^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.0792032355106864195506908480\dots$ Stable Faltings height: $0.41310098042308445758078235482\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $7.0144143340292381702111744635\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.12481281137931516112895358729\dots$ 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: $8$  = $2\cdot2^{2}$ 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: $1$ (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: $L'(E,1)$ ≈ $1.7509775464191117511001905894892692301$

## 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);

$$q - 2 q^{3} + q^{9} + 4 q^{13} - 6 q^{17} + 2 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 6912 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 2 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$(j)_{-}$
$2$ $2$ $I_{13}^{*}$ Additive -1 4 21 9
$7$ $4$ $I_{2}^{*}$ Additive -1 2 8 2

## 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 8.6.0.6
$3$ 3B 9.12.0.1

## $p$-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.

## 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 ordinary ss add ss ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary - 3 1,3 - 1,1 1 1 1 1,1 3 1 1 1 1 1 - 2 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, 3, 6, 9 and 18.
Its isogeny class 784.b consists of 6 curves linked by isogenies of degrees dividing 18.

## 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 $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{-21})$$ $$\Z/6\Z$$ Not in database $4$ 4.0.392.1 $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{-21})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.2.784147392.1 $$\Z/6\Z$$ Not in database $6$ 6.0.21171979584.2 $$\Z/18\Z$$ Not in database $8$ 8.0.9834496.2 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.4.40282095616.3 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.796594176.9 $$\Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/18\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ 12.0.1836043139093761282277376.5 $$\Z/2\Z \times \Z/18\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ 16.0.162447943996702457856.3 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ 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.