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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 3234.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3234.i1 | 3234o3 | \([1, 0, 1, -4508222, -3684683824]\) | \(86129359107301290313/9166294368\) | \(1078405366100832\) | \([2]\) | \(92160\) | \(2.3123\) | |
| 3234.i2 | 3234o2 | \([1, 0, 1, -282462, -57291440]\) | \(21184262604460873/216872764416\) | \(25514863860777984\) | \([2, 2]\) | \(46080\) | \(1.9657\) | |
| 3234.i3 | 3234o4 | \([1, 0, 1, -70782, -141116720]\) | \(-333345918055753/72923718045024\) | \(-8579402504279028576\) | \([2]\) | \(92160\) | \(2.3123\) | |
| 3234.i4 | 3234o1 | \([1, 0, 1, -31582, 712016]\) | \(29609739866953/15259926528\) | \(1795315096092672\) | \([2]\) | \(23040\) | \(1.6191\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3234.i have rank \(0\).
Complex multiplication
The elliptic curves in class 3234.i do not have complex multiplication.Modular form 3234.2.a.i
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.
