# Properties

 Label 504c Number of curves $4$ Conductor $504$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

## Elliptic curves in class 504c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
504.c4 504c1 [0, 0, 0, 9, -54]  64 $$\Gamma_0(N)$$-optimal
504.c3 504c2 [0, 0, 0, -171, -810] [2, 2] 128
504.c1 504c3 [0, 0, 0, -2691, -53730]  256
504.c2 504c4 [0, 0, 0, -531, 3726]  256

## Rank

sage: E.rank()

The elliptic curves in class 504c have rank $$0$$.

## Complex multiplication

The elliptic curves in class 504c do not have complex multiplication.

## Modular form504.2.a.c

sage: E.q_eigenform(10)

$$q - 2q^{5} - q^{7} + 4q^{11} + 2q^{13} + 6q^{17} + 8q^{19} + O(q^{20})$$

## 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. 