# Properties

 Label 770.d Number of curves $4$ Conductor $770$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 770.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
770.d1 770e3 $$[1, -1, 0, -20909, -1158507]$$ $$1010962818911303721/57392720$$ $$57392720$$ $$$$ $$1024$$ $$0.95539$$
770.d2 770e4 $$[1, -1, 0, -2189, 9845]$$ $$1160306142246441/634128110000$$ $$634128110000$$ $$$$ $$1024$$ $$0.95539$$
770.d3 770e2 $$[1, -1, 0, -1309, -17787]$$ $$248158561089321/1859334400$$ $$1859334400$$ $$[2, 2]$$ $$512$$ $$0.60882$$
770.d4 770e1 $$[1, -1, 0, -29, -635]$$ $$-2749884201/176619520$$ $$-176619520$$ $$$$ $$256$$ $$0.26225$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 770.d do not have complex multiplication.

## Modular form770.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - 3 q^{9} - q^{10} - q^{11} - 6 q^{13} - q^{14} + q^{16} - 2 q^{17} + 3 q^{18} - 4 q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 