# Properties

 Label 41971a Number of curves $3$ Conductor $41971$ CM no Rank $0$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 41971a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
41971.a3 41971a1 $$[0, 1, 1, 1473, 1452]$$ $$32768/19$$ $$-204805091251$$ $$[]$$ $$35190$$ $$0.85990$$ $$\Gamma_0(N)$$-optimal
41971.a2 41971a2 $$[0, 1, 1, -20617, 1205357]$$ $$-89915392/6859$$ $$-73934637941611$$ $$[]$$ $$105570$$ $$1.4092$$
41971.a1 41971a3 $$[0, 1, 1, -1699457, 852167382]$$ $$-50357871050752/19$$ $$-204805091251$$ $$[]$$ $$316710$$ $$1.9585$$

## Rank

sage: E.rank()

The elliptic curves in class 41971a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 41971a do not have complex multiplication.

## Modular form 41971.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{3} - 2q^{4} - 3q^{5} - q^{7} + q^{9} - 3q^{11} + 4q^{12} + 4q^{13} + 6q^{15} + 4q^{16} - 3q^{17} - q^{19} + O(q^{20})$$ ## 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 Cremona numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 