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SageMath
E = EllipticCurve("da1")
E.isogeny_class()
Elliptic curves in class 87360.da
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 87360.da1 | 87360bh7 | \([0, -1, 0, -145003425, -671911569375]\) | \(1286229821345376481036009/247265484375000000\) | \(64819163136000000000000\) | \([2]\) | \(15925248\) | \(3.3776\) | |
| 87360.da2 | 87360bh8 | \([0, -1, 0, -63779745, 189919770657]\) | \(109454124781830273937129/3914078300576808000\) | \(1026052142026406756352000\) | \([2]\) | \(15925248\) | \(3.3776\) | |
| 87360.da3 | 87360bh5 | \([0, -1, 0, -63221985, 193507413345]\) | \(106607603143751752938169/5290068420\) | \(1386759695892480\) | \([2]\) | \(5308416\) | \(2.8283\) | |
| 87360.da4 | 87360bh6 | \([0, -1, 0, -10019745, -8142821343]\) | \(424378956393532177129/136231857216000000\) | \(35712363978031104000000\) | \([2, 2]\) | \(7962624\) | \(3.0311\) | |
| 87360.da5 | 87360bh4 | \([0, -1, 0, -4400865, 2294301537]\) | \(35958207000163259449/12145729518877500\) | \(3183930118996623360000\) | \([2]\) | \(5308416\) | \(2.8283\) | |
| 87360.da6 | 87360bh2 | \([0, -1, 0, -3951585, 3024201825]\) | \(26031421522845051769/5797789779600\) | \(1519855803983462400\) | \([2, 2]\) | \(2654208\) | \(2.4818\) | |
| 87360.da7 | 87360bh1 | \([0, -1, 0, -219105, 58373217]\) | \(-4437543642183289/3033210136320\) | \(-795137837975470080\) | \([2]\) | \(1327104\) | \(2.1352\) | \(\Gamma_0(N)\)-optimal |
| 87360.da8 | 87360bh3 | \([0, -1, 0, 1776735, -869111775]\) | \(2366200373628880151/2612420149248000\) | \(-684830267604467712000\) | \([2]\) | \(3981312\) | \(2.6845\) |
Rank
sage: E.rank()
The elliptic curves in class 87360.da have rank \(1\).
Complex multiplication
The elliptic curves in class 87360.da do not have complex multiplication.Modular form 87360.2.a.da
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.
