# Properties

 Label 458640nl Number of curves $4$ Conductor $458640$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("nl1")

sage: E.isogeny_class()

## Elliptic curves in class 458640nl

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.nl4 458640nl1 $$[0, 0, 0, -8967, -1160026]$$ $$-3631696/24375$$ $$-535180595040000$$ $$[2]$$ $$1769472$$ $$1.5095$$ $$\Gamma_0(N)$$-optimal*
458640.nl3 458640nl2 $$[0, 0, 0, -229467, -42217126]$$ $$15214885924/38025$$ $$3339526913049600$$ $$[2, 2]$$ $$3538944$$ $$1.8561$$ $$\Gamma_0(N)$$-optimal*
458640.nl2 458640nl3 $$[0, 0, 0, -317667, -6778366]$$ $$20183398562/11567205$$ $$2031768173899376640$$ $$[2]$$ $$7077888$$ $$2.2026$$ $$\Gamma_0(N)$$-optimal*
458640.nl1 458640nl4 $$[0, 0, 0, -3669267, -2705310286]$$ $$31103978031362/195$$ $$34251558082560$$ $$[2]$$ $$7077888$$ $$2.2026$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 458640nl1.

## Rank

sage: E.rank()

The elliptic curves in class 458640nl have rank $$0$$.

## Complex multiplication

The elliptic curves in class 458640nl do not have complex multiplication.

## Modular form 458640.2.a.nl

sage: E.q_eigenform(10)

$$q + q^{5} + 4q^{11} - q^{13} + 6q^{17} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.