# Properties

 Label 70699.a Number of curves $3$ Conductor $70699$ CM no Rank $0$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 70699.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
70699.a1 70699a3 $$[0, 1, 1, -2862689, -1865226945]$$ $$-50357871050752/19$$ $$-978887112859$$ $$[]$$ $$691740$$ $$2.0889$$
70699.a2 70699a2 $$[0, 1, 1, -34729, -2661790]$$ $$-89915392/6859$$ $$-353378247742099$$ $$[]$$ $$230580$$ $$1.5396$$
70699.a3 70699a1 $$[0, 1, 1, 2481, -1275]$$ $$32768/19$$ $$-978887112859$$ $$[]$$ $$76860$$ $$0.99026$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 70699.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 70699.a do not have complex multiplication.

## Modular form 70699.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{3} - 2q^{4} + 3q^{5} + q^{7} + q^{9} - 3q^{11} + 4q^{12} - 4q^{13} - 6q^{15} + 4q^{16} + 3q^{17} + q^{19} + O(q^{20})$$ ## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels. 