# Properties

 Label 768.d Number of curves $4$ Conductor $768$ CM no Rank $0$ Graph # Related objects

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

E.isogeny_class()

## Elliptic curves in class 768.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
768.d1 768f3 $$[0, -1, 0, -647, 6555]$$ $$58591911104/243$$ $$124416$$ $$$$ $$160$$ $$0.18700$$
768.d2 768f4 $$[0, -1, 0, -637, 6757]$$ $$-873722816/59049$$ $$-1934917632$$ $$$$ $$320$$ $$0.53358$$
768.d3 768f1 $$[0, -1, 0, -7, -5]$$ $$85184/3$$ $$1536$$ $$$$ $$32$$ $$-0.61772$$ $$\Gamma_0(N)$$-optimal
768.d4 768f2 $$[0, -1, 0, 3, -27]$$ $$64/9$$ $$-294912$$ $$$$ $$64$$ $$-0.27114$$

## Rank

sage: E.rank()

The elliptic curves in class 768.d have rank $$0$$.

## Complex multiplication

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

## Modular form768.2.a.d

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. 