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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 40950.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40950.r1 | 40950r7 | \([1, -1, 0, -509777667, 4429550638741]\) | \(1286229821345376481036009/247265484375000000\) | \(2816508407958984375000000\) | \([2]\) | \(15925248\) | \(3.6920\) | |
| 40950.r2 | 40950r8 | \([1, -1, 0, -224225667, -1251716353259]\) | \(109454124781830273937129/3914078300576808000\) | \(44583798142507703625000000\) | \([2]\) | \(15925248\) | \(3.6920\) | |
| 40950.r3 | 40950r5 | \([1, -1, 0, -222264792, -1275367080884]\) | \(106607603143751752938169/5290068420\) | \(60257185596562500\) | \([2]\) | \(5308416\) | \(3.1426\) | |
| 40950.r4 | 40950r6 | \([1, -1, 0, -35225667, 53706646741]\) | \(424378956393532177129/136231857216000000\) | \(1551765998601000000000000\) | \([2, 2]\) | \(7962624\) | \(3.3454\) | |
| 40950.r5 | 40950r4 | \([1, -1, 0, -15471792, -15110031884]\) | \(35958207000163259449/12145729518877500\) | \(138347450300964023437500\) | \([2]\) | \(5308416\) | \(3.1426\) | |
| 40950.r6 | 40950r2 | \([1, -1, 0, -13892292, -19922768384]\) | \(26031421522845051769/5797789779600\) | \(66040449208256250000\) | \([2, 2]\) | \(2654208\) | \(2.7961\) | |
| 40950.r7 | 40950r1 | \([1, -1, 0, -770292, -384110384]\) | \(-4437543642183289/3033210136320\) | \(-34550159209020000000\) | \([2]\) | \(1327104\) | \(2.4495\) | \(\Gamma_0(N)\)-optimal |
| 40950.r8 | 40950r3 | \([1, -1, 0, 6246333, 5723542741]\) | \(2366200373628880151/2612420149248000\) | \(-29757098262528000000000\) | \([2]\) | \(3981312\) | \(2.9988\) |
Rank
sage: E.rank()
The elliptic curves in class 40950.r have rank \(0\).
Complex multiplication
The elliptic curves in class 40950.r do not have complex multiplication.Modular form 40950.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
