# Properties

 Label 504.c Number of curves $4$ Conductor $504$ CM no Rank $0$ Graph # Learn more

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

sage: E.isogeny_class()

## Elliptic curves in class 504.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
504.c1 504c3 $$[0, 0, 0, -2691, -53730]$$ $$1443468546/7$$ $$10450944$$ $$$$ $$256$$ $$0.54727$$
504.c2 504c4 $$[0, 0, 0, -531, 3726]$$ $$11090466/2401$$ $$3584673792$$ $$$$ $$256$$ $$0.54727$$
504.c3 504c2 $$[0, 0, 0, -171, -810]$$ $$740772/49$$ $$36578304$$ $$[2, 2]$$ $$128$$ $$0.20070$$
504.c4 504c1 $$[0, 0, 0, 9, -54]$$ $$432/7$$ $$-1306368$$ $$$$ $$64$$ $$-0.14587$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 504.c 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 