# Properties

 Label 53040a Number of curves $2$ Conductor $53040$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 53040a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53040.m1 53040a1 $$[0, -1, 0, -281016, -57158784]$$ $$2396726313900986596/4154072495625$$ $$4253770235520000$$ $$$$ $$368640$$ $$1.8928$$ $$\Gamma_0(N)$$-optimal
53040.m2 53040a2 $$[0, -1, 0, -193136, -93646560]$$ $$-389032340685029858/1627263833203125$$ $$-3332636330400000000$$ $$$$ $$737280$$ $$2.2393$$

## Rank

sage: E.rank()

The elliptic curves in class 53040a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 53040a do not have complex multiplication.

## Modular form 53040.2.a.a

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{9} - 2 q^{11} - q^{13} + q^{15} - q^{17} + 8 q^{19} + O(q^{20})$$

## 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 Cremona 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 Cremona labels. 