# Properties

 Label 468270bf Number of curves $4$ Conductor $468270$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bf1")

sage: E.isogeny_class()

## Elliptic curves in class 468270bf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
468270.bf3 468270bf1 [1, -1, 0, -74619, -7817067] [2] 2211840 $$\Gamma_0(N)$$-optimal*
468270.bf2 468270bf2 [1, -1, 0, -96399, -2864295] [2, 2] 4423680 $$\Gamma_0(N)$$-optimal*
468270.bf1 468270bf3 [1, -1, 0, -913149, 333800055] [2] 8847360 $$\Gamma_0(N)$$-optimal*
468270.bf4 468270bf4 [1, -1, 0, 371871, -22812597] [2] 8847360
*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 468270bf1.

## Rank

sage: E.rank()

The elliptic curves in class 468270bf have rank $$1$$.

## Complex multiplication

The elliptic curves in class 468270bf do not have complex multiplication.

## Modular form 468270.2.a.bf

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{5} - 4q^{7} - q^{8} - q^{10} + 2q^{13} + 4q^{14} + q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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 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.