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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 342.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 342.e1 | 342a3 | \([1, -1, 1, -770, 66305]\) | \(-69173457625/2550136832\) | \(-1859049750528\) | \([3]\) | \(540\) | \(1.0345\) | |
| 342.e2 | 342a1 | \([1, -1, 1, -140, -601]\) | \(-413493625/152\) | \(-110808\) | \([]\) | \(60\) | \(-0.064137\) | \(\Gamma_0(N)\)-optimal |
| 342.e3 | 342a2 | \([1, -1, 1, 85, -2437]\) | \(94196375/3511808\) | \(-2560108032\) | \([3]\) | \(180\) | \(0.48517\) |
Rank
sage: E.rank()
The elliptic curves in class 342.e have rank \(0\).
Complex multiplication
The elliptic curves in class 342.e do not have complex multiplication.Modular form 342.2.a.e
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
