# Properties

 Label 53312bk Number of curves $2$ Conductor $53312$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("53312.h1")

sage: E.isogeny_class()

## Elliptic curves in class 53312bk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53312.h1 53312bk1 [0, 1, 0, -14177, -740321] [] 161280 $$\Gamma_0(N)$$-optimal
53312.h2 53312bk2 [0, 1, 0, 95583, 2881759] [] 483840

## Rank

sage: E.rank()

The elliptic curves in class 53312bk have rank $$0$$.

## Modular form 53312.2.a.h

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 Cremona 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 Cremona labels. 