# Properties

 Label 458640na Number of curves $6$ Conductor $458640$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("458640.na1")

sage: E.isogeny_class()

## Elliptic curves in class 458640na

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
458640.na6 458640na1 [0, 0, 0, 105693, -5296606] [2] 4718592 $$\Gamma_0(N)$$-optimal*
458640.na5 458640na2 [0, 0, 0, -458787, -44019934] [2, 2] 9437184 $$\Gamma_0(N)$$-optimal*
458640.na3 458640na3 [0, 0, 0, -3986787, 3033101666] [2, 2] 18874368 $$\Gamma_0(N)$$-optimal*
458640.na2 458640na4 [0, 0, 0, -5962467, -5599434526] [2] 18874368
458640.na1 458640na5 [0, 0, 0, -63609987, 195270223106] [2] 37748736 $$\Gamma_0(N)$$-optimal*
458640.na4 458640na6 [0, 0, 0, -811587, 7731762626] [2] 37748736
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 458640na1.

## Rank

sage: E.rank()

The elliptic curves in class 458640na have rank $$1$$.

## Modular form 458640.2.a.na

sage: E.q_eigenform(10)

$$q + q^{5} + 4q^{11} - q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.