# Properties

 Label 53312o Number of curves $4$ Conductor $53312$ CM no Rank $2$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("53312.ba1")

sage: E.isogeny_class()

## Elliptic curves in class 53312o

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53312.ba3 53312o1 [0, 0, 0, -2156, -16464] [2] 36864 $$\Gamma_0(N)$$-optimal
53312.ba2 53312o2 [0, 0, 0, -17836, 905520] [2, 2] 73728
53312.ba4 53312o3 [0, 0, 0, -2156, 2442160] [2] 147456
53312.ba1 53312o4 [0, 0, 0, -284396, 58375856] [2] 147456

## Rank

sage: E.rank()

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

## Modular form 53312.2.a.ba

sage: E.q_eigenform(10)

$$q - 2q^{5} - 3q^{9} - 2q^{13} - q^{17} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.