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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 176610u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 176610.cy7 | 176610u1 | \([1, 1, 1, -418415, 103959605]\) | \(13619385906841/6048000\) | \(3597491445408000\) | \([2]\) | \(2322432\) | \(1.9435\) | \(\Gamma_0(N)\)-optimal |
| 176610.cy6 | 176610u2 | \([1, 1, 1, -485695, 68193557]\) | \(21302308926361/8930250000\) | \(5311920962360250000\) | \([2, 2]\) | \(4644864\) | \(2.2900\) | |
| 176610.cy5 | 176610u3 | \([1, 1, 1, -1238390, -403642765]\) | \(353108405631241/86318776320\) | \(51344421195318558720\) | \([2]\) | \(6967296\) | \(2.4928\) | |
| 176610.cy8 | 176610u4 | \([1, 1, 1, 1616805, 502990557]\) | \(785793873833639/637994920500\) | \(-379494257392940980500\) | \([2]\) | \(9289728\) | \(2.6366\) | |
| 176610.cy4 | 176610u5 | \([1, 1, 1, -3664675, -2654284915]\) | \(9150443179640281/184570312500\) | \(109786726239257812500\) | \([2]\) | \(9289728\) | \(2.6366\) | |
| 176610.cy2 | 176610u6 | \([1, 1, 1, -18462070, -30538193293]\) | \(1169975873419524361/108425318400\) | \(64493907971170406400\) | \([2, 2]\) | \(13934592\) | \(2.8393\) | |
| 176610.cy3 | 176610u7 | \([1, 1, 1, -17116470, -35176745613]\) | \(-932348627918877961/358766164249920\) | \(-213402481281568888384320\) | \([2]\) | \(27869184\) | \(3.1859\) | |
| 176610.cy1 | 176610u8 | \([1, 1, 1, -295386550, -1954166401165]\) | \(4791901410190533590281/41160000\) | \(24482927892360000\) | \([2]\) | \(27869184\) | \(3.1859\) |
Rank
sage: E.rank()
The elliptic curves in class 176610u have rank \(0\).
Complex multiplication
The elliptic curves in class 176610u do not have complex multiplication.Modular form 176610.2.a.u
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.
