# Properties

 Label 72.a Number of curves $6$ Conductor $72$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 72.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
72.a1 72a5 $$[0, 0, 0, -3459, 78302]$$ $$3065617154/9$$ $$13436928$$ $$$$ $$32$$ $$0.59710$$
72.a2 72a3 $$[0, 0, 0, -579, -5362]$$ $$28756228/3$$ $$2239488$$ $$$$ $$16$$ $$0.25053$$
72.a3 72a4 $$[0, 0, 0, -219, 1190]$$ $$1556068/81$$ $$60466176$$ $$[2, 2]$$ $$16$$ $$0.25053$$
72.a4 72a2 $$[0, 0, 0, -39, -70]$$ $$35152/9$$ $$1679616$$ $$[2, 2]$$ $$8$$ $$-0.096046$$
72.a5 72a1 $$[0, 0, 0, 6, -7]$$ $$2048/3$$ $$-34992$$ $$$$ $$4$$ $$-0.44262$$ $$\Gamma_0(N)$$-optimal
72.a6 72a6 $$[0, 0, 0, 141, 4718]$$ $$207646/6561$$ $$-9795520512$$ $$$$ $$32$$ $$0.59710$$

## Rank

sage: E.rank()

The elliptic curves in class 72.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 72.a do not have complex multiplication.

## Modular form72.2.a.a

sage: E.q_eigenform(10)

$$q + 2q^{5} - 4q^{11} - 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$ ## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 