# Properties

 Label 53312s Number of curves $2$ Conductor $53312$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("53312.bc1")

sage: E.isogeny_class()

## Elliptic curves in class 53312s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53312.bc2 53312s1 [0, 0, 0, 5684, -82320] [2] 73728 $$\Gamma_0(N)$$-optimal
53312.bc1 53312s2 [0, 0, 0, -25676, -696976] [2] 147456

## Rank

sage: E.rank()

The elliptic curves in class 53312s have rank $$1$$.

## Modular form 53312.2.a.bc

sage: E.q_eigenform(10)

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

## 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.