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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 1386.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1386.g1 | 1386g3 | \([1, -1, 1, -828041, -289811415]\) | \(86129359107301290313/9166294368\) | \(6682228594272\) | \([2]\) | \(15360\) | \(1.8886\) | |
| 1386.g2 | 1386g2 | \([1, -1, 1, -51881, -4494999]\) | \(21184262604460873/216872764416\) | \(158100245259264\) | \([2, 2]\) | \(7680\) | \(1.5420\) | |
| 1386.g3 | 1386g4 | \([1, -1, 1, -13001, -11104599]\) | \(-333345918055753/72923718045024\) | \(-53161390454822496\) | \([2]\) | \(15360\) | \(1.8886\) | |
| 1386.g4 | 1386g1 | \([1, -1, 1, -5801, 57705]\) | \(29609739866953/15259926528\) | \(11124486438912\) | \([4]\) | \(3840\) | \(1.1955\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1386.g have rank \(1\).
Complex multiplication
The elliptic curves in class 1386.g do not have complex multiplication.Modular form 1386.2.a.g
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
