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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 39039.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 39039.r1 | 39039h6 | \([1, 1, 0, -763714, -257206607]\) | \(10206027697760497/5557167\) | \(26823383690103\) | \([2]\) | \(307200\) | \(1.9053\) | |
| 39039.r2 | 39039h4 | \([1, 1, 0, -47999, -3986640]\) | \(2533811507137/58110129\) | \(280486493648361\) | \([2, 2]\) | \(153600\) | \(1.5587\) | |
| 39039.r3 | 39039h2 | \([1, 1, 0, -6594, 112455]\) | \(6570725617/2614689\) | \(12620604397401\) | \([2, 2]\) | \(76800\) | \(1.2121\) | |
| 39039.r4 | 39039h1 | \([1, 1, 0, -5749, 165352]\) | \(4354703137/1617\) | \(7804950153\) | \([2]\) | \(38400\) | \(0.86554\) | \(\Gamma_0(N)\)-optimal |
| 39039.r5 | 39039h5 | \([1, 1, 0, 5236, -12280653]\) | \(3288008303/13504609503\) | \(-65184170690565927\) | \([2]\) | \(307200\) | \(1.9053\) | |
| 39039.r6 | 39039h3 | \([1, 1, 0, 21291, 843042]\) | \(221115865823/190238433\) | \(-918244580550297\) | \([2]\) | \(153600\) | \(1.5587\) |
Rank
sage: E.rank()
The elliptic curves in class 39039.r have rank \(0\).
Complex multiplication
The elliptic curves in class 39039.r do not have complex multiplication.Modular form 39039.2.a.r
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.
