# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("53312.g1")

sage: E.isogeny_class()

## Elliptic curves in class 53312bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53312.g2 53312bd1 [0, 1, 0, -188225, -3750881] [2] 1032192 $$\Gamma_0(N)$$-optimal
53312.g1 53312bd2 [0, 1, 0, -2195265, -1250122721] [2] 2064384

## Rank

sage: E.rank()

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

## Modular form 53312.2.a.g

sage: E.q_eigenform(10)

$$q - 2q^{3} - 4q^{5} + q^{9} + 6q^{11} - 2q^{13} + 8q^{15} + q^{17} + O(q^{20})$$

## 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.