Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 4140.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4140.i1 | 4140h2 | \([0, 0, 0, -12432, 539044]\) | \(-1138621087744/13687875\) | \(-2554485984000\) | \([3]\) | \(6912\) | \(1.1925\) | |
4140.i2 | 4140h1 | \([0, 0, 0, 528, 3796]\) | \(87228416/83835\) | \(-15645623040\) | \([]\) | \(2304\) | \(0.64321\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4140.i have rank \(1\).
Complex multiplication
The elliptic curves in class 4140.i do not have complex multiplication.Modular form 4140.2.a.i
sage: E.q_eigenform(10)
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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.