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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 34848.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34848.i1 | 34848cg4 | \([0, 0, 0, -35211, -2542210]\) | \(7301384/3\) | \(1983694800384\) | \([2]\) | \(81920\) | \(1.3220\) | |
34848.i2 | 34848cg3 | \([0, 0, 0, -18876, 979616]\) | \(140608/3\) | \(15869558403072\) | \([2]\) | \(81920\) | \(1.3220\) | |
34848.i3 | 34848cg1 | \([0, 0, 0, -2541, -26620]\) | \(21952/9\) | \(743885550144\) | \([2, 2]\) | \(40960\) | \(0.97546\) | \(\Gamma_0(N)\)-optimal |
34848.i4 | 34848cg2 | \([0, 0, 0, 8349, -194326]\) | \(97336/81\) | \(-53559759610368\) | \([2]\) | \(81920\) | \(1.3220\) |
Rank
sage: E.rank()
The elliptic curves in class 34848.i have rank \(2\).
Complex multiplication
The elliptic curves in class 34848.i do not have complex multiplication.Modular form 34848.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.