Properties

 Label 333270n Number of curves $8$ Conductor $333270$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("333270.n1")

sage: E.isogeny_class()

Elliptic curves in class 333270n

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
333270.n7 333270n1 [1, -1, 0, -195300, -11640240] [2] 4866048 $$\Gamma_0(N)$$-optimal
333270.n5 333270n2 [1, -1, 0, -1718820, 859508496] [2, 2] 9732096
333270.n4 333270n3 [1, -1, 0, -12764340, -17549564544] [2] 14598144
333270.n2 333270n4 [1, -1, 0, -27428220, 55296592056] [2] 19464192
333270.n6 333270n5 [1, -1, 0, -385740, 2157661800] [2] 19464192
333270.n3 333270n6 [1, -1, 0, -12859560, -17274359700] [2, 2] 29196288
333270.n1 333270n7 [1, -1, 0, -30713310, 41225237550] [2] 58392576
333270.n8 333270n8 [1, -1, 0, 3470670, -58161989574] [2] 58392576

Rank

sage: E.rank()

The elliptic curves in class 333270n have rank $$1$$.

Modular form 333270.2.a.n

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + q^{10} + 2q^{13} + q^{14} + q^{16} - 6q^{17} + 4q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels.