# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("53312.q1")

sage: E.isogeny_class()

## Elliptic curves in class 53312.q

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53312.q1 53312bc1 [0, 1, 0, -289, -2241] [] 23040 $$\Gamma_0(N)$$-optimal
53312.q2 53312bc2 [0, 1, 0, 1951, 8959] [] 69120

## Rank

sage: E.rank()

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

## Modular form 53312.2.a.q

sage: E.q_eigenform(10)

$$q - 2q^{3} + 3q^{5} + q^{9} + 2q^{13} - 6q^{15} + q^{17} - 7q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.