# Properties

 Label 141610s Number of curves $4$ Conductor $141610$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("141610.cg1")

sage: E.isogeny_class()

## Elliptic curves in class 141610s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
141610.cg4 141610s1 [1, -1, 1, 32747, 3625437]  983040 $$\Gamma_0(N)$$-optimal
141610.cg3 141610s2 [1, -1, 1, -250473, 39084581] [2, 2] 1966080
141610.cg1 141610s3 [1, -1, 1, -3790723, 2841546481]  3932160
141610.cg2 141610s4 [1, -1, 1, -1241743, -497390743]  3932160

## Rank

sage: E.rank()

The elliptic curves in class 141610s have rank $$1$$.

## Modular form 141610.2.a.cg

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{5} + q^{8} - 3q^{9} - q^{10} - 4q^{11} + 6q^{13} + q^{16} - 3q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 