# Properties

 Label 770f Number of curves 4 Conductor 770 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("770.f1")

sage: E.isogeny_class()

## Elliptic curves in class 770f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
770.f3 770f1 [1, 0, 0, -56, 3136]  576 $$\Gamma_0(N)$$-optimal
770.f2 770f2 [1, 0, 0, -3576, 81280]  1152
770.f4 770f3 [1, 0, 0, 504, -84560]  1728
770.f1 770f4 [1, 0, 0, -26116, -1580604]  3456

## Rank

sage: E.rank()

The elliptic curves in class 770f have rank $$1$$.

## Modular form770.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - q^{11} - 2q^{12} - 4q^{13} + q^{14} + 2q^{15} + q^{16} + q^{18} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 