# Properties

 Label 402930dj Number of curves $6$ Conductor $402930$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("402930.dj1")

sage: E.isogeny_class()

## Elliptic curves in class 402930dj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
402930.dj5 402930dj1 [1, -1, 1, -920228, 1169881287] [2] 17694720 $$\Gamma_0(N)$$-optimal*
402930.dj4 402930dj2 [1, -1, 1, -23222948, 42991941831] [2, 2] 35389440 $$\Gamma_0(N)$$-optimal*
402930.dj1 402930dj3 [1, -1, 1, -371354468, 2754518724807] [2] 70778880 $$\Gamma_0(N)$$-optimal*
402930.dj3 402930dj4 [1, -1, 1, -31934948, 7809401031] [2, 2] 70778880
402930.dj6 402930dj5 [1, -1, 1, 126841252, 62174371911] [2] 141557760
402930.dj2 402930dj6 [1, -1, 1, -330103148, -2298342725049] [2] 141557760
*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 402930dj1.

## Rank

sage: E.rank()

The elliptic curves in class 402930dj have rank $$1$$.

## Modular form 402930.2.a.dj

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} + 2q^{13} + q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.