# Properties

 Label 13860.r Number of curves $2$ Conductor $13860$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("r1")

sage: E.isogeny_class()

## Elliptic curves in class 13860.r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13860.r1 13860e1 $$[0, 0, 0, -309312, -66212991]$$ $$10392086293512192/1684375$$ $$530456850000$$ $$$$ $$74880$$ $$1.6517$$ $$\Gamma_0(N)$$-optimal
13860.r2 13860e2 $$[0, 0, 0, -308367, -66637674]$$ $$-643570518871152/8271484375$$ $$-41678752500000000$$ $$$$ $$149760$$ $$1.9983$$

## Rank

sage: E.rank()

The elliptic curves in class 13860.r have rank $$0$$.

## Complex multiplication

The elliptic curves in class 13860.r do not have complex multiplication.

## Modular form 13860.2.a.r

sage: E.q_eigenform(10)

$$q + q^{5} - q^{7} + q^{11} + 6q^{13} + 6q^{17} + O(q^{20})$$ ## 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. 