# Properties

 Label 10780e Number of curves $2$ Conductor $10780$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 10780e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10780.e2 10780e1 $$[0, -1, 0, 124, -440]$$ $$16674224/15125$$ $$-189728000$$ $$[]$$ $$3456$$ $$0.27576$$ $$\Gamma_0(N)$$-optimal
10780.e1 10780e2 $$[0, -1, 0, -1276, 24200]$$ $$-18330740176/8857805$$ $$-111112305920$$ $$[]$$ $$10368$$ $$0.82507$$

## Rank

sage: E.rank()

The elliptic curves in class 10780e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 10780e do not have complex multiplication.

## Modular form 10780.2.a.e

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} - 2q^{9} - q^{11} + 4q^{13} + q^{15} - 6q^{17} - 2q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 