# Properties

 Label 768.a Number of curves $4$ Conductor $768$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("a1")

E.isogeny_class()

## Elliptic curves in class 768.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
768.a1 768b4 $$[0, -1, 0, -2589, -49851]$$ $$58591911104/243$$ $$7962624$$ $$$$ $$320$$ $$0.53358$$
768.a2 768b3 $$[0, -1, 0, -159, -765]$$ $$-873722816/59049$$ $$-30233088$$ $$$$ $$160$$ $$0.18700$$
768.a3 768b2 $$[0, -1, 0, -29, 69]$$ $$85184/3$$ $$98304$$ $$$$ $$64$$ $$-0.27114$$
768.a4 768b1 $$[0, -1, 0, 1, 3]$$ $$64/9$$ $$-4608$$ $$$$ $$32$$ $$-0.61772$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 768.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 768.a do not have complex multiplication.

## Modular form768.2.a.a

sage: E.q_eigenform(10)

$$q - q^{3} - 2 q^{5} + 2 q^{7} + q^{9} - 4 q^{13} + 2 q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 