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SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 283920dl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 283920.dl7 | 283920dl1 | \([0, -1, 0, -9257200, -16007601728]\) | \(-4437543642183289/3033210136320\) | \(-59968413634070949396480\) | \([2]\) | \(27869184\) | \(3.0711\) | \(\Gamma_0(N)\)-optimal |
| 283920.dl6 | 283920dl2 | \([0, -1, 0, -166954480, -830104040000]\) | \(26031421522845051769/5797789779600\) | \(114625838646400190054400\) | \([2, 2]\) | \(55738368\) | \(3.4177\) | |
| 283920.dl8 | 283920dl3 | \([0, -1, 0, 75067040, 238492153600]\) | \(2366200373628880151/2612420149248000\) | \(-51649139049150831132672000\) | \([2]\) | \(83607552\) | \(3.6204\) | |
| 283920.dl5 | 283920dl4 | \([0, -1, 0, -185936560, -629607718208]\) | \(35958207000163259449/12145729518877500\) | \(240128477402249571932160000\) | \([4]\) | \(111476736\) | \(3.7642\) | |
| 283920.dl3 | 283920dl5 | \([0, -1, 0, -2671128880, -53135295567680]\) | \(106607603143751752938169/5290068420\) | \(104587877827673210880\) | \([2]\) | \(111476736\) | \(3.7642\) | |
| 283920.dl4 | 283920dl6 | \([0, -1, 0, -423334240, 2237280646912]\) | \(424378956393532177129/136231857216000000\) | \(2693386872819317735424000000\) | \([2, 2]\) | \(167215104\) | \(3.9670\) | |
| 283920.dl1 | 283920dl7 | \([0, -1, 0, -6126394720, 184539030726400]\) | \(1286229821345376481036009/247265484375000000\) | \(4888589374958016000000000000\) | \([4]\) | \(334430208\) | \(4.3135\) | |
| 283920.dl2 | 283920dl8 | \([0, -1, 0, -2694694240, -52149980281088]\) | \(109454124781830273937129/3914078300576808000\) | \(77383714275036537019072512000\) | \([2]\) | \(334430208\) | \(4.3135\) |
Rank
sage: E.rank()
The elliptic curves in class 283920dl have rank \(0\).
Complex multiplication
The elliptic curves in class 283920dl do not have complex multiplication.Modular form 283920.2.a.dl
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
