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SageMath
E = EllipticCurve("fy1")
E.isogeny_class()
Elliptic curves in class 109200fy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
109200.gk7 | 109200fy1 | \([0, 1, 0, -1369408, -911396812]\) | \(-4437543642183289/3033210136320\) | \(-194125448724480000000\) | \([2]\) | \(3981312\) | \(2.5933\) | \(\Gamma_0(N)\)-optimal |
109200.gk6 | 109200fy2 | \([0, 1, 0, -24697408, -47240804812]\) | \(26031421522845051769/5797789779600\) | \(371058545894400000000\) | \([2, 2]\) | \(7962624\) | \(2.9399\) | |
109200.gk8 | 109200fy3 | \([0, 1, 0, 11104592, 13574319188]\) | \(2366200373628880151/2612420149248000\) | \(-167194889551872000000000\) | \([2]\) | \(11943936\) | \(3.1426\) | |
109200.gk5 | 109200fy4 | \([0, 1, 0, -27505408, -35834708812]\) | \(35958207000163259449/12145729518877500\) | \(777326689208160000000000\) | \([2]\) | \(15925248\) | \(3.2865\) | |
109200.gk3 | 109200fy5 | \([0, 1, 0, -395137408, -3023355764812]\) | \(106607603143751752938169/5290068420\) | \(338564378880000000\) | \([2]\) | \(15925248\) | \(3.2865\) | |
109200.gk4 | 109200fy6 | \([0, 1, 0, -62623408, 127262895188]\) | \(424378956393532177129/136231857216000000\) | \(8718838861824000000000000\) | \([2, 2]\) | \(23887872\) | \(3.4892\) | |
109200.gk1 | 109200fy7 | \([0, 1, 0, -906271408, 10499071407188]\) | \(1286229821345376481036009/247265484375000000\) | \(15824991000000000000000000\) | \([2]\) | \(47775744\) | \(3.8358\) | |
109200.gk2 | 109200fy8 | \([0, 1, 0, -398623408, -2967297104812]\) | \(109454124781830273937129/3914078300576808000\) | \(250501011236915712000000000\) | \([2]\) | \(47775744\) | \(3.8358\) |
Rank
sage: E.rank()
The elliptic curves in class 109200fy have rank \(0\).
Complex multiplication
The elliptic curves in class 109200fy do not have complex multiplication.Modular form 109200.2.a.fy
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.