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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 83391.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 83391.p1 | 83391r6 | \([1, 0, 1, -1631367, 801865639]\) | \(10206027697760497/5557167\) | \(261441817379127\) | \([2]\) | \(1105920\) | \(2.0950\) | |
| 83391.p2 | 83391r4 | \([1, 0, 1, -102532, 12375245]\) | \(2533811507137/58110129\) | \(2733842213828649\) | \([2, 2]\) | \(552960\) | \(1.7484\) | |
| 83391.p3 | 83391r2 | \([1, 0, 1, -14087, -360835]\) | \(6570725617/2614689\) | \(123010347546009\) | \([2, 2]\) | \(276480\) | \(1.4019\) | |
| 83391.p4 | 83391r1 | \([1, 0, 1, -12282, -524729]\) | \(4354703137/1617\) | \(76073189577\) | \([2]\) | \(138240\) | \(1.0553\) | \(\Gamma_0(N)\)-optimal |
| 83391.p5 | 83391r5 | \([1, 0, 1, 11183, 38347751]\) | \(3288008303/13504609503\) | \(-635336251629607143\) | \([2]\) | \(1105920\) | \(2.0950\) | |
| 83391.p6 | 83391r3 | \([1, 0, 1, 45478, -2600479]\) | \(221115865823/190238433\) | \(-8949934680544473\) | \([2]\) | \(552960\) | \(1.7484\) |
Rank
sage: E.rank()
The elliptic curves in class 83391.p have rank \(2\).
Complex multiplication
The elliptic curves in class 83391.p do not have complex multiplication.Modular form 83391.2.a.p
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
