# Properties

 Label 283920ej Number of curves $6$ Conductor $283920$ CM no Rank $2$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("283920.ej1")

sage: E.isogeny_class()

## Elliptic curves in class 283920ej

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
283920.ej4 283920ej1 [0, 1, 0, -29631, -1973100] [2] 589824 $$\Gamma_0(N)$$-optimal
283920.ej3 283920ej2 [0, 1, 0, -30476, -1855476] [2, 2] 1179648
283920.ej2 283920ej3 [0, 1, 0, -114976, 12982724] [2, 2] 2359296
283920.ej5 283920ej4 [0, 1, 0, 40504, -9152220] [2] 2359296
283920.ej1 283920ej5 [0, 1, 0, -1771176, 906668244] [2] 4718592
283920.ej6 283920ej6 [0, 1, 0, 189224, 70294004] [2] 4718592

## Rank

sage: E.rank()

The elliptic curves in class 283920ej have rank $$2$$.

## Modular form 283920.2.a.ej

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} - q^{7} + q^{9} - 4q^{11} - q^{15} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.