Properties

 Label 1288f Number of curves $2$ Conductor $1288$ CM no Rank $1$ Graph Related objects

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

Elliptic curves in class 1288f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1288.c2 1288f1 $$[0, 1, 0, 4192, 16816]$$ $$7953970437500/4703287687$$ $$-4816166591488$$ $$$$ $$1920$$ $$1.1229$$ $$\Gamma_0(N)$$-optimal
1288.c1 1288f2 $$[0, 1, 0, -16968, 118384]$$ $$263822189935250/149429406721$$ $$306031424964608$$ $$$$ $$3840$$ $$1.4695$$

Rank

sage: E.rank()

The elliptic curves in class 1288f have rank $$1$$.

Complex multiplication

The elliptic curves in class 1288f do not have complex multiplication.

Modular form1288.2.a.f

sage: E.q_eigenform(10)

$$q - 2 q^{3} + q^{7} + q^{9} - 6 q^{17} + 6 q^{19} + O(q^{20})$$ 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels. 