# Properties

 Label 53312.cc Number of curves 4 Conductor 53312 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("53312.cc1")

sage: E.isogeny_class()

## Elliptic curves in class 53312.cc

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53312.cc1 53312cb4 [0, -1, 0, -354433, -56005599] [2] 663552
53312.cc2 53312cb3 [0, -1, 0, -323073, -70562911] [2] 331776
53312.cc3 53312cb2 [0, -1, 0, -134913, 19114145] [2] 221184
53312.cc4 53312cb1 [0, -1, 0, -9473, 222881] [2] 110592 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form 53312.2.a.cc

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.