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SageMath
E = EllipticCurve("ig1")
E.isogeny_class()
Elliptic curves in class 278850ig
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 278850.ig6 | 278850ig1 | \([1, 0, 0, 1077287, 40991417]\) | \(1833318007919/1070530560\) | \(-80738227215360000000\) | \([2]\) | \(8847360\) | \(2.5092\) | \(\Gamma_0(N)\)-optimal |
| 278850.ig5 | 278850ig2 | \([1, 0, 0, -4330713, 327615417]\) | \(119102750067601/68309049600\) | \(5151792740480100000000\) | \([2, 2]\) | \(17694720\) | \(2.8558\) | |
| 278850.ig2 | 278850ig3 | \([1, 0, 0, -49960713, 135620565417]\) | \(182864522286982801/463015182960\) | \(34920091441374603750000\) | \([2]\) | \(35389440\) | \(3.2024\) | |
| 278850.ig3 | 278850ig4 | \([1, 0, 0, -45228713, -116599766583]\) | \(135670761487282321/643043610000\) | \(48497635689695156250000\) | \([2, 2]\) | \(35389440\) | \(3.2024\) | |
| 278850.ig4 | 278850ig5 | \([1, 0, 0, -21991213, -236249654083]\) | \(-15595206456730321/310672490129100\) | \(-23430574553242985029687500\) | \([2]\) | \(70778880\) | \(3.5490\) | |
| 278850.ig1 | 278850ig6 | \([1, 0, 0, -722834213, -7480138735083]\) | \(553808571467029327441/12529687500\) | \(944975131127929687500\) | \([2]\) | \(70778880\) | \(3.5490\) |
Rank
sage: E.rank()
The elliptic curves in class 278850ig have rank \(1\).
Complex multiplication
The elliptic curves in class 278850ig do not have complex multiplication.Modular form 278850.2.a.ig
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
