# Properties

 Label 259182.ct Number of curves $2$ Conductor $259182$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("ct1")

sage: E.isogeny_class()

## Elliptic curves in class 259182.ct

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
259182.ct1 259182ct2 $$[1, -1, 0, -1241961, -532423171]$$ $$8086832279405361/226576$$ $$5935854588048$$ $$$$ $$2949120$$ $$1.9617$$
259182.ct2 259182ct1 $$[1, -1, 0, -77721, -8282323]$$ $$1981858514481/10449152$$ $$273747646884096$$ $$$$ $$1474560$$ $$1.6151$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 259182.ct have rank $$1$$.

## Complex multiplication

The elliptic curves in class 259182.ct do not have complex multiplication.

## Modular form 259182.2.a.ct

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + 2 q^{5} + q^{7} - q^{8} - 2 q^{10} + 4 q^{13} - q^{14} + q^{16} + q^{17} - 6 q^{19} + O(q^{20})$$ ## 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. 