Properties

 Label 287490ca Number of curves $8$ Conductor $287490$ CM no Rank $0$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("287490.ca1")

sage: E.isogeny_class()

Elliptic curves in class 287490ca

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
287490.ca7 287490ca1 [1, 0, 1, -56158, -1807024] [2] 2488320 $$\Gamma_0(N)$$-optimal
287490.ca5 287490ca2 [1, 0, 1, -494238, 132420688] [2, 2] 4976640
287490.ca4 287490ca3 [1, 0, 1, -3670318, -2706775792] [2] 7464960
287490.ca2 287490ca4 [1, 0, 1, -7886838, 8524500208] [2] 9953280
287490.ca6 287490ca5 [1, 0, 1, -110918, 332667056] [2] 9953280
287490.ca3 287490ca6 [1, 0, 1, -3697698, -2664347744] [2, 2] 14929920
287490.ca1 287490ca7 [1, 0, 1, -8831448, 6354624256] [2] 29859840
287490.ca8 287490ca8 [1, 0, 1, 997972, -8967815152] [2] 29859840

Rank

sage: E.rank()

The elliptic curves in class 287490ca have rank $$0$$.

Modular form 287490.2.a.ca

sage: E.q_eigenform(10)

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.