# Properties

 Label 13860.f Number of curves $4$ Conductor $13860$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 13860.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13860.f1 13860o4 $$[0, 0, 0, -1765263, 901827862]$$ $$3259751350395879376/3806353980275$$ $$710357005214841600$$ $$$$ $$248832$$ $$2.3375$$
13860.f2 13860o3 $$[0, 0, 0, -1764768, 902359393]$$ $$52112158467655991296/71177645$$ $$830216051280$$ $$$$ $$124416$$ $$1.9909$$
13860.f3 13860o2 $$[0, 0, 0, -82263, -7803938]$$ $$329890530231376/49933296875$$ $$9318751596000000$$ $$$$ $$82944$$ $$1.7882$$
13860.f4 13860o1 $$[0, 0, 0, -22368, 1168333]$$ $$106110329552896/10850811125$$ $$126563860962000$$ $$$$ $$41472$$ $$1.4416$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 13860.2.a.f

sage: E.q_eigenform(10)

$$q - q^{5} + q^{7} - q^{11} - 4q^{13} - 4q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 