# Properties

 Label 471510ep Number of curves $6$ Conductor $471510$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("471510.ep1")

sage: E.isogeny_class()

## Elliptic curves in class 471510ep

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
471510.ep6 471510ep1 [1, -1, 1, 91228, 60067271] [2] 7864320 $$\Gamma_0(N)$$-optimal*
471510.ep5 471510ep2 [1, -1, 1, -1855652, 919809479] [2, 2] 15728640 $$\Gamma_0(N)$$-optimal*
471510.ep2 471510ep3 [1, -1, 1, -29233652, 60844775879] [2, 2] 31457280 $$\Gamma_0(N)$$-optimal*
471510.ep4 471510ep4 [1, -1, 1, -5627732, -4005018169] [2] 31457280
471510.ep1 471510ep5 [1, -1, 1, -467737952, 3893723161319] [2] 62914560 $$\Gamma_0(N)$$-optimal*
471510.ep3 471510ep6 [1, -1, 1, -28777352, 62835704039] [2] 62914560
*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 471510ep1.

## Rank

sage: E.rank()

The elliptic curves in class 471510ep have rank $$1$$.

## Modular form 471510.2.a.ep

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} - 4q^{11} + q^{16} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.