# Properties

 Label 388311d Number of curves 6 Conductor 388311 CM no Rank 1 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("388311.d1")

sage: E.isogeny_class()

## Elliptic curves in class 388311d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
388311.d4 388311d1 [1, 0, 0, -57189, 5257560] [2] 1331200 $$\Gamma_0(N)$$-optimal
388311.d3 388311d2 [1, 0, 0, -65594, 3608499] [2, 2] 2662400
388311.d6 388311d3 [1, 0, 0, 211771, 26186010] [2] 5324800
388311.d2 388311d4 [1, 0, 0, -477439, -124475296] [2, 2] 5324800
388311.d5 388311d5 [1, 0, 0, 52076, -385314385] [2] 10649600
388311.d1 388311d6 [1, 0, 0, -7596474, -8059351707] [2] 10649600

## Rank

sage: E.rank()

The elliptic curves in class 388311d have rank $$1$$.

## Modular form 388311.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} - q^{7} + 3q^{8} + q^{9} + 2q^{10} + q^{11} - q^{12} - 6q^{13} + q^{14} - 2q^{15} - q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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 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.