# Properties

 Label 53312.k Number of curves 4 Conductor 53312 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("53312.k1")

sage: E.isogeny_class()

## Elliptic curves in class 53312.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53312.k1 53312ba4 [0, 1, 0, -354433, 56005599]  663552
53312.k2 53312ba3 [0, 1, 0, -323073, 70562911]  331776
53312.k3 53312ba2 [0, 1, 0, -134913, -19114145]  221184
53312.k4 53312ba1 [0, 1, 0, -9473, -222881]  110592 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form 53312.2.a.k

sage: E.q_eigenform(10)

$$q - 2q^{3} + q^{9} - 6q^{11} + 2q^{13} + q^{17} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 