# Properties

 Label 53312.m Number of curves $2$ Conductor $53312$ CM no Rank $2$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("53312.m1")

sage: E.isogeny_class()

## Elliptic curves in class 53312.m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53312.m1 53312cg2 [0, 1, 0, -9473, 336895]  98304
53312.m2 53312cg1 [0, 1, 0, -1633, -19041]  49152 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 53312.m have rank $$2$$.

## Modular form 53312.2.a.m

sage: E.q_eigenform(10)

$$q - 2q^{3} + q^{9} + 2q^{11} - 6q^{13} + q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 