Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 1369.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1369.c1 | 1369a3 | \([0, 1, 1, -2564593, -1581651042]\) | \(727057727488000/37\) | \(94931877133\) | \([]\) | \(8208\) | \(2.0275\) | |
1369.c2 | 1369a2 | \([0, 1, 1, -31943, -2138543]\) | \(1404928000/50653\) | \(129961739795077\) | \([]\) | \(2736\) | \(1.4782\) | |
1369.c3 | 1369a1 | \([0, 1, 1, -4563, 116200]\) | \(4096000/37\) | \(94931877133\) | \([]\) | \(912\) | \(0.92893\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1369.c have rank \(1\).
Complex multiplication
The elliptic curves in class 1369.c do not have complex multiplication.Modular form 1369.2.a.c
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.