Properties

 Label 3648q Number of curves $2$ Conductor $3648$ CM no Rank $1$ Graph Related objects

Show commands: SageMath
sage: E = EllipticCurve("q1")

sage: E.isogeny_class()

Elliptic curves in class 3648q

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3648.v2 3648q1 $$[0, 1, 0, 11, 155]$$ $$131072/9747$$ $$-9980928$$ $$$$ $$576$$ $$0.022556$$ $$\Gamma_0(N)$$-optimal
3648.v1 3648q2 $$[0, 1, 0, -369, 2511]$$ $$340062928/13851$$ $$226934784$$ $$$$ $$1152$$ $$0.36913$$

Rank

sage: E.rank()

The elliptic curves in class 3648q have rank $$1$$.

Complex multiplication

The elliptic curves in class 3648q do not have complex multiplication.

Modular form3648.2.a.q

sage: E.q_eigenform(10)

$$q + q^{3} - 2 q^{5} + q^{9} - 2 q^{11} - 2 q^{13} - 2 q^{15} + 6 q^{17} + q^{19} + O(q^{20})$$ 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. 