# Properties

 Label 422370dw Number of curves $6$ Conductor $422370$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("422370.dw1")

sage: E.isogeny_class()

## Elliptic curves in class 422370dw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
422370.dw6 422370dw1 [1, -1, 1, 48667, 1642781] [2] 3538944 $$\Gamma_0(N)$$-optimal*
422370.dw5 422370dw2 [1, -1, 1, -211253, 13807037] [2, 2] 7077888 $$\Gamma_0(N)$$-optimal*
422370.dw2 422370dw3 [1, -1, 1, -2745473, 1750254581] [2] 14155776 $$\Gamma_0(N)$$-optimal*
422370.dw3 422370dw4 [1, -1, 1, -1835753, -947247163] [2, 2] 14155776
422370.dw4 422370dw5 [1, -1, 1, -373703, -2415730183] [2] 28311552
422370.dw1 422370dw6 [1, -1, 1, -29289803, -61005726943] [2] 28311552
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 422370dw1.

## Rank

sage: E.rank()

The elliptic curves in class 422370dw have rank $$0$$.

## Modular form 422370.2.a.dw

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} - 4q^{11} - q^{13} + q^{16} + 6q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.