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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1848.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1848.d1 | 1848b3 | \([0, -1, 0, -640232, -139597332]\) | \(14171198121996897746/4077720290568771\) | \(8351171155084843008\) | \([2]\) | \(46080\) | \(2.3371\) | |
| 1848.d2 | 1848b2 | \([0, -1, 0, -586992, -172882980]\) | \(21843440425782779332/3100814593569\) | \(3175234143814656\) | \([2, 2]\) | \(23040\) | \(1.9906\) | |
| 1848.d3 | 1848b1 | \([0, -1, 0, -586972, -172895372]\) | \(87364831012240243408/1760913\) | \(450793728\) | \([2]\) | \(11520\) | \(1.6440\) | \(\Gamma_0(N)\)-optimal |
| 1848.d4 | 1848b4 | \([0, -1, 0, -534072, -205375860]\) | \(-8226100326647904626/4152140742401883\) | \(-8503584240439056384\) | \([2]\) | \(46080\) | \(2.3371\) |
Rank
sage: E.rank()
The elliptic curves in class 1848.d have rank \(1\).
Complex multiplication
The elliptic curves in class 1848.d do not have complex multiplication.Modular form 1848.2.a.d
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.
