# Properties

 Label 735.c Number of curves 8 Conductor 735 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("735.c1")

sage: E.isogeny_class()

## Elliptic curves in class 735.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
735.c1 735e7 [1, 0, 0, -105841, 13244636] [2] 1536
735.c2 735e5 [1, 0, 0, -6616, 206471] [2, 2] 768
735.c3 735e8 [1, 0, 0, -5391, 285606] [2] 1536
735.c4 735e3 [1, 0, 0, -3921, -94830] [2] 384
735.c5 735e4 [1, 0, 0, -491, 1896] [2, 2] 384
735.c6 735e2 [1, 0, 0, -246, -1485] [2, 2] 192
735.c7 735e1 [1, 0, 0, -1, -64] [2] 96 $$\Gamma_0(N)$$-optimal
735.c8 735e6 [1, 0, 0, 1714, 14685] [2] 768

## Rank

sage: E.rank()

The elliptic curves in class 735.c have rank $$1$$.

## Modular form735.2.a.c

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + 3q^{8} + q^{9} + q^{10} - 4q^{11} - q^{12} + 2q^{13} - q^{15} - q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.