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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 34496cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34496.bn1 | 34496cp1 | \([0, -1, 0, -17509, 897611]\) | \(-78843215872/539\) | \(-4058419904\) | \([]\) | \(46080\) | \(1.0245\) | \(\Gamma_0(N)\)-optimal |
| 34496.bn2 | 34496cp2 | \([0, -1, 0, -9669, 1695331]\) | \(-13278380032/156590819\) | \(-1179056208929984\) | \([]\) | \(138240\) | \(1.5738\) | |
| 34496.bn3 | 34496cp3 | \([0, -1, 0, 86371, -43827629]\) | \(9463555063808/115539436859\) | \(-869958349249567424\) | \([]\) | \(414720\) | \(2.1231\) |
Rank
sage: E.rank()
The elliptic curves in class 34496cp have rank \(1\).
Complex multiplication
The elliptic curves in class 34496cp do not have complex multiplication.Modular form 34496.2.a.cp
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
