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SageMath
E = EllipticCurve("fj1")
E.isogeny_class()
Elliptic curves in class 52416fj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 52416.fc3 | 52416fj1 | \([0, 0, 0, -4044, -88688]\) | \(38272753/4368\) | \(834737799168\) | \([2]\) | \(73728\) | \(1.0199\) | \(\Gamma_0(N)\)-optimal |
| 52416.fc2 | 52416fj2 | \([0, 0, 0, -15564, 653200]\) | \(2181825073/298116\) | \(56970854793216\) | \([2, 2]\) | \(147456\) | \(1.3664\) | |
| 52416.fc4 | 52416fj3 | \([0, 0, 0, 24756, 3475600]\) | \(8780064047/32388174\) | \(-6189476438605824\) | \([2]\) | \(294912\) | \(1.7130\) | |
| 52416.fc1 | 52416fj4 | \([0, 0, 0, -240204, 45311632]\) | \(8020417344913/187278\) | \(35789383139328\) | \([2]\) | \(294912\) | \(1.7130\) |
Rank
sage: E.rank()
The elliptic curves in class 52416fj have rank \(0\).
Complex multiplication
The elliptic curves in class 52416fj do not have complex multiplication.Modular form 52416.2.a.fj
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
