# Properties

 Label 35.a Number of curves $3$ Conductor $35$ CM no Rank $0$ Graph # Learn more

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

sage: E.isogeny_class()

## Elliptic curves in class 35.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
35.a1 35a2 $$[0, 1, 1, -131, -650]$$ $$-250523582464/13671875$$ $$-13671875$$ $$[]$$ $$6$$ $$0.12746$$
35.a2 35a3 $$[0, 1, 1, -1, 0]$$ $$-262144/35$$ $$-35$$ $$$$ $$6$$ $$-0.97115$$
35.a3 35a1 $$[0, 1, 1, 9, 1]$$ $$71991296/42875$$ $$-42875$$ $$$$ $$2$$ $$-0.42184$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 35.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 35.a do not have complex multiplication.

## Modular form35.2.a.a

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 