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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 3234n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3234.n4 | 3234n1 | \([1, 0, 1, 170, 536]\) | \(4657463/3696\) | \(-434830704\) | \([2]\) | \(1536\) | \(0.34547\) | \(\Gamma_0(N)\)-optimal |
| 3234.n3 | 3234n2 | \([1, 0, 1, -810, 4456]\) | \(498677257/213444\) | \(25111473156\) | \([2, 2]\) | \(3072\) | \(0.69204\) | |
| 3234.n2 | 3234n3 | \([1, 0, 1, -6200, -185272]\) | \(223980311017/4278582\) | \(503370893718\) | \([2]\) | \(6144\) | \(1.0386\) | |
| 3234.n1 | 3234n4 | \([1, 0, 1, -11100, 448984]\) | \(1285429208617/614922\) | \(72344958378\) | \([2]\) | \(6144\) | \(1.0386\) |
Rank
sage: E.rank()
The elliptic curves in class 3234n have rank \(0\).
Complex multiplication
The elliptic curves in class 3234n do not have complex multiplication.Modular form 3234.2.a.n
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.
