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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 19110j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19110.m7 | 19110j1 | \([1, 1, 0, -167752, -39126464]\) | \(-4437543642183289/3033210136320\) | \(-356854139327911680\) | \([2]\) | \(331776\) | \(2.0684\) | \(\Gamma_0(N)\)-optimal |
| 19110.m6 | 19110j2 | \([1, 1, 0, -3025432, -2026357136]\) | \(26031421522845051769/5797789779600\) | \(682104169780160400\) | \([2, 2]\) | \(663552\) | \(2.4150\) | |
| 19110.m8 | 19110j3 | \([1, 1, 0, 1360313, 582407029]\) | \(2366200373628880151/2612420149248000\) | \(-307348618138877952000\) | \([2]\) | \(995328\) | \(2.6177\) | |
| 19110.m3 | 19110j4 | \([1, 1, 0, -48404332, -129640899716]\) | \(106607603143751752938169/5290068420\) | \(622371259544580\) | \([2]\) | \(1327104\) | \(2.7616\) | |
| 19110.m5 | 19110j5 | \([1, 1, 0, -3369412, -1537423964]\) | \(35958207000163259449/12145729518877500\) | \(1428932932166418997500\) | \([2]\) | \(1327104\) | \(2.7616\) | |
| 19110.m4 | 19110j6 | \([1, 1, 0, -7671367, 5454095221]\) | \(424378956393532177129/136231857216000000\) | \(16027541769605184000000\) | \([2, 2]\) | \(1990656\) | \(2.9643\) | |
| 19110.m2 | 19110j7 | \([1, 1, 0, -48831367, -127237512779]\) | \(109454124781830273937129/3914078300576808000\) | \(460487397984560884392000\) | \([2]\) | \(3981312\) | \(3.3109\) | |
| 19110.m1 | 19110j8 | \([1, 1, 0, -111018247, 450114381109]\) | \(1286229821345376481036009/247265484375000000\) | \(29090536971234375000000\) | \([2]\) | \(3981312\) | \(3.3109\) |
Rank
sage: E.rank()
The elliptic curves in class 19110j have rank \(1\).
Complex multiplication
The elliptic curves in class 19110j do not have complex multiplication.Modular form 19110.2.a.j
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.
