# Properties

 Label 52.a Number of curves $2$ Conductor $52$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 52.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
52.a1 52a2 $$[0, 0, 0, -4, -3]$$ $$442368/13$$ $$208$$ $$$$ $$6$$ $$-0.77787$$
52.a2 52a1 $$[0, 0, 0, 1, -10]$$ $$432/169$$ $$-43264$$ $$$$ $$3$$ $$-0.43130$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 52.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 52.a do not have complex multiplication.

## Modular form52.2.a.a

sage: E.q_eigenform(10)

$$q + 2 q^{5} - 2 q^{7} - 3 q^{9} - 2 q^{11} - q^{13} + 6 q^{17} - 6 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 