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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 15680.ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 15680.ca1 | 15680dj3 | \([0, 0, 0, -2561132, -1577596944]\) | \(481927184300808/1225\) | \(4722524979200\) | \([2]\) | \(147456\) | \(2.0956\) | |
| 15680.ca2 | 15680dj4 | \([0, 0, 0, -209132, -8303344]\) | \(262389836808/144120025\) | \(555600341277900800\) | \([2]\) | \(147456\) | \(2.0956\) | |
| 15680.ca3 | 15680dj2 | \([0, 0, 0, -160132, -24630144]\) | \(942344950464/1500625\) | \(723136637440000\) | \([2, 2]\) | \(73728\) | \(1.7490\) | |
| 15680.ca4 | 15680dj1 | \([0, 0, 0, -7007, -620144]\) | \(-5053029696/19140625\) | \(-144120025000000\) | \([2]\) | \(36864\) | \(1.4024\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 15680.ca have rank \(0\).
Complex multiplication
The elliptic curves in class 15680.ca do not have complex multiplication.Modular form 15680.2.a.ca
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
