# Properties

 Label 35a Number of curves 3 Conductor 35 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("35.a1")

sage: E.isogeny_class()

## Elliptic curves in class 35a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
35.a3 35a1 [0, 1, 1, 9, 1]  2 $$\Gamma_0(N)$$-optimal
35.a1 35a2 [0, 1, 1, -131, -650] [] 6
35.a2 35a3 [0, 1, 1, -1, 0]  6

## Rank

sage: E.rank()

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 