# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 13860x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13860.x3 13860x1 $$[0, 0, 0, -23952, -1447931]$$ $$-130287139815424/2250652635$$ $$-26251612334640$$ $$$$ $$41472$$ $$1.3728$$ $$\Gamma_0(N)$$-optimal
13860.x2 13860x2 $$[0, 0, 0, -384807, -91878194]$$ $$33766427105425744/9823275$$ $$1833258873600$$ $$$$ $$82944$$ $$1.7194$$
13860.x4 13860x3 $$[0, 0, 0, 92688, -6938759]$$ $$7549996227362816/6152409907875$$ $$-71761709165454000$$ $$$$ $$124416$$ $$1.9222$$
13860.x1 13860x4 $$[0, 0, 0, -446367, -60520826]$$ $$52702650535889104/22020583921875$$ $$4109569453836000000$$ $$$$ $$248832$$ $$2.2687$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 13860.2.a.x

sage: E.q_eigenform(10)

$$q + q^{5} + q^{7} + q^{11} + 2q^{13} + 6q^{17} + 8q^{19} + O(q^{20})$$ ## 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 Cremona 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 Cremona labels. 