# Properties

 Label 735.f Number of curves 4 Conductor 735 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("735.f1")

sage: E.isogeny_class()

## Elliptic curves in class 735.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
735.f1 735a3 [1, 1, 0, -5513, 155268] [2] 768
735.f2 735a2 [1, 1, 0, -368, 1947] [2, 2] 384
735.f3 735a1 [1, 1, 0, -123, -552] [2] 192 $$\Gamma_0(N)$$-optimal
735.f4 735a4 [1, 1, 0, 857, 13462] [2] 768

## Rank

sage: E.rank()

The elliptic curves in class 735.f have rank $$0$$.

## Modular form735.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 3q^{8} + q^{9} - q^{10} + q^{12} + 6q^{13} + q^{15} - q^{16} - 2q^{17} + q^{18} + 8q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.