# Properties

 Label 458640.ld Number of curves 8 Conductor 458640 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("458640.ld1")

sage: E.isogeny_class()

## Elliptic curves in class 458640.ld

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
458640.ld1 458640ld7 [0, 0, 0, -283583850627, 58125989146260994] [2] 1019215872 $$\Gamma_0(N)$$-optimal*
458640.ld2 458640ld6 [0, 0, 0, -17724001827, 908217379192354] [2, 2] 509607936 $$\Gamma_0(N)$$-optimal*
458640.ld3 458640ld8 [0, 0, 0, -17506183107, 931627794634306] [2] 1019215872
458640.ld4 458640ld4 [0, 0, 0, -3502633827, 79657407097954] [2] 339738624 $$\Gamma_0(N)$$-optimal*
458640.ld5 458640ld3 [0, 0, 0, -1121374947, 13823907590626] [2] 254803968 $$\Gamma_0(N)$$-optimal*
458640.ld6 458640ld2 [0, 0, 0, -292153827, 339930313954] [2, 2] 169869312 $$\Gamma_0(N)$$-optimal*
458640.ld7 458640ld1 [0, 0, 0, -181515747, -936235683614] [2] 84934656 $$\Gamma_0(N)$$-optimal*
458640.ld8 458640ld5 [0, 0, 0, 1148116893, 2697077374306] [2] 339738624
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 6 curves highlighted, and conditionally curve 458640.ld7.

## Rank

sage: E.rank()

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

## Modular form 458640.2.a.ld

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.