# Properties

 Label 5610.e Number of curves $2$ Conductor $5610$ CM no Rank $1$ Graph # Related objects

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

E.isogeny_class()

## Elliptic curves in class 5610.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5610.e1 5610e1 $$[1, 1, 0, -88, -308]$$ $$76711450249/12622500$$ $$12622500$$ $$$$ $$1920$$ $$0.084170$$ $$\Gamma_0(N)$$-optimal
5610.e2 5610e2 $$[1, 1, 0, 162, -1458]$$ $$465664585751/1274620050$$ $$-1274620050$$ $$$$ $$3840$$ $$0.43074$$

## Rank

sage: E.rank()

The elliptic curves in class 5610.e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 5610.e do not have complex multiplication.

## Modular form5610.2.a.e

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} + q^{10} + q^{11} - q^{12} - 4 q^{13} - 4 q^{14} + q^{15} + q^{16} + q^{17} - q^{18} + 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 