# Properties

 Label 491970.ci Number of curves $6$ Conductor $491970$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("491970.ci1")

sage: E.isogeny_class()

## Elliptic curves in class 491970.ci

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
491970.ci1 491970ci5 [1, 0, 0, -162678091, 798609261845] [2] 46137344 $$\Gamma_0(N)$$-optimal*
491970.ci2 491970ci3 [1, 0, 0, -10167391, 12477607625] [2, 2] 23068672 $$\Gamma_0(N)$$-optimal*
491970.ci3 491970ci6 [1, 0, 0, -10008691, 12886006205] [2] 46137344
491970.ci4 491970ci4 [1, 0, 0, -1957311, -821926359] [2] 23068672
491970.ci5 491970ci2 [1, 0, 0, -645391, 188514425] [2, 2] 11534336 $$\Gamma_0(N)$$-optimal*
491970.ci6 491970ci1 [1, 0, 0, 31729, 12327801] [2] 5767168 $$\Gamma_0(N)$$-optimal*
*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 491970.ci6.

## Rank

sage: E.rank()

The elliptic curves in class 491970.ci have rank $$0$$.

## Modular form 491970.2.a.ci

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} + 4q^{11} + q^{12} + 6q^{13} - q^{15} + q^{16} - 2q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.