Show commands:
SageMath
E = EllipticCurve("di1")
E.isogeny_class()
Elliptic curves in class 121680di
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121680.x1 | 121680di1 | \([0, 0, 0, -1443, 22178]\) | \(-658489/40\) | \(-20185251840\) | \([]\) | \(82944\) | \(0.73224\) | \(\Gamma_0(N)\)-optimal |
121680.x2 | 121680di2 | \([0, 0, 0, 7917, 35282]\) | \(108750551/64000\) | \(-32296402944000\) | \([]\) | \(248832\) | \(1.2815\) |
Rank
sage: E.rank()
The elliptic curves in class 121680di have rank \(1\).
Complex multiplication
The elliptic curves in class 121680di do not have complex multiplication.Modular form 121680.2.a.di
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 Cremona 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 Cremona labels.