# Properties

 Label 78.a Number of curves $4$ Conductor $78$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 78.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
78.a1 78a4 $$[1, 1, 0, -20739, 1140957]$$ $$986551739719628473/111045168$$ $$111045168$$ $$$$ $$160$$ $$0.96808$$
78.a2 78a3 $$[1, 1, 0, -2339, -15747]$$ $$1416134368422073/725251155408$$ $$725251155408$$ $$$$ $$160$$ $$0.96808$$
78.a3 78a2 $$[1, 1, 0, -1299, 17325]$$ $$242702053576633/2554695936$$ $$2554695936$$ $$[2, 2]$$ $$80$$ $$0.62150$$
78.a4 78a1 $$[1, 1, 0, -19, 685]$$ $$-822656953/207028224$$ $$-207028224$$ $$$$ $$40$$ $$0.27493$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 78.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 78.a do not have complex multiplication.

## Modular form78.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} + q^{13} - 4 q^{14} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - 8 q^{19} + O(q^{20})$$

## 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. 