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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 3312n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3312.h3 | 3312n1 | \([0, 0, 0, -5115, 147242]\) | \(-4956477625/268272\) | \(-801055899648\) | \([2]\) | \(3072\) | \(1.0426\) | \(\Gamma_0(N)\)-optimal |
| 3312.h2 | 3312n2 | \([0, 0, 0, -82875, 9182954]\) | \(21081759765625/57132\) | \(170595237888\) | \([2]\) | \(6144\) | \(1.3892\) | |
| 3312.h4 | 3312n3 | \([0, 0, 0, 27285, 301466]\) | \(752329532375/448524288\) | \(-1339286347579392\) | \([2]\) | \(9216\) | \(1.5919\) | |
| 3312.h1 | 3312n4 | \([0, 0, 0, -110955, 2430362]\) | \(50591419971625/28422890688\) | \(84870296828116992\) | \([2]\) | \(18432\) | \(1.9385\) |
Rank
sage: E.rank()
The elliptic curves in class 3312n have rank \(1\).
Complex multiplication
The elliptic curves in class 3312n do not have complex multiplication.Modular form 3312.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
