# Properties

 Label 10304.l Number of curves $2$ Conductor $10304$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("l1")

sage: E.isogeny_class()

## Elliptic curves in class 10304.l

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10304.l1 10304bg2 $$[0, 1, 0, -11137, -455553]$$ $$582810602977/829472$$ $$217441107968$$ $$$$ $$15360$$ $$1.0784$$
10304.l2 10304bg1 $$[0, 1, 0, -897, -2945]$$ $$304821217/164864$$ $$43218108416$$ $$$$ $$7680$$ $$0.73180$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 10304.l have rank $$1$$.

## Complex multiplication

The elliptic curves in class 10304.l do not have complex multiplication.

## Modular form 10304.2.a.l

sage: E.q_eigenform(10)

$$q - 2q^{3} + 2q^{5} + q^{7} + q^{9} - 2q^{11} + 4q^{13} - 4q^{15} - 6q^{17} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 