# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("eu1")

sage: E.isogeny_class()

## Elliptic curves in class 259182.eu

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
259182.eu1 259182eu2 $$[1, -1, 1, -30515, 253905]$$ $$2433138625/1387778$$ $$1792270835082882$$ $$$$ $$1075200$$ $$1.6166$$
259182.eu2 259182eu1 $$[1, -1, 1, -19625, -1048539]$$ $$647214625/3332$$ $$4303171272708$$ $$$$ $$537600$$ $$1.2701$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 259182.2.a.eu

sage: E.q_eigenform(10)

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