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SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 35490dl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 35490.dm7 | 35490dl1 | \([1, 0, 0, -578575, 250118777]\) | \(-4437543642183289/3033210136320\) | \(-14640725984880602880\) | \([4]\) | \(1161216\) | \(2.3779\) | \(\Gamma_0(N)\)-optimal |
| 35490.dm6 | 35490dl2 | \([1, 0, 0, -10434655, 12970375625]\) | \(26031421522845051769/5797789779600\) | \(27984823888281296400\) | \([2, 2]\) | \(2322432\) | \(2.7245\) | |
| 35490.dm8 | 35490dl3 | \([1, 0, 0, 4691690, -3726439900]\) | \(2366200373628880151/2612420149248000\) | \(-12609653088171589632000\) | \([4]\) | \(3483648\) | \(2.9273\) | |
| 35490.dm5 | 35490dl4 | \([1, 0, 0, -11621035, 9837620597]\) | \(35958207000163259449/12145729518877500\) | \(58625116553283586897500\) | \([2]\) | \(4644864\) | \(3.0711\) | |
| 35490.dm3 | 35490dl5 | \([1, 0, 0, -166945555, 830238993245]\) | \(106607603143751752938169/5290068420\) | \(25534149860271780\) | \([2]\) | \(4644864\) | \(3.0711\) | |
| 35490.dm4 | 35490dl6 | \([1, 0, 0, -26458390, -34957510108]\) | \(424378956393532177129/136231857216000000\) | \(657565154496903744000000\) | \([2, 2]\) | \(6967296\) | \(3.2738\) | |
| 35490.dm1 | 35490dl7 | \([1, 0, 0, -382899670, -2883422355100]\) | \(1286229821345376481036009/247265484375000000\) | \(1193503265370609375000000\) | \([2]\) | \(13934592\) | \(3.6204\) | |
| 35490.dm2 | 35490dl8 | \([1, 0, 0, -168418390, 814843441892]\) | \(109454124781830273937129/3914078300576808000\) | \(18892508367928842045672000\) | \([2]\) | \(13934592\) | \(3.6204\) |
Rank
sage: E.rank()
The elliptic curves in class 35490dl have rank \(0\).
Complex multiplication
The elliptic curves in class 35490dl do not have complex multiplication.Modular form 35490.2.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.
