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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 46410cn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 46410.ck7 | 46410cn1 | \([1, 0, 0, -298010, 293238372]\) | \(-2926956820564562516641/35459588343029760000\) | \(-35459588343029760000\) | \([8]\) | \(1572864\) | \(2.4331\) | \(\Gamma_0(N)\)-optimal |
| 46410.ck6 | 46410cn2 | \([1, 0, 0, -8696090, 9838496100]\) | \(72727020009972527154752161/265361167808100000000\) | \(265361167808100000000\) | \([2, 8]\) | \(3145728\) | \(2.7796\) | |
| 46410.ck5 | 46410cn3 | \([1, 0, 0, -12746090, -254913900]\) | \(229010110533436633465952161/132501160769452503210000\) | \(132501160769452503210000\) | \([2, 4]\) | \(6291456\) | \(3.1262\) | |
| 46410.ck3 | 46410cn4 | \([1, 0, 0, -139015370, 630861993012]\) | \(297106512928238351998640242081/3977028808593750000\) | \(3977028808593750000\) | \([8]\) | \(6291456\) | \(3.1262\) | |
| 46410.ck8 | 46410cn5 | \([1, 0, 0, 50978410, -2026455000]\) | \(14651516183052242700771495839/8480668142378708755560900\) | \(-8480668142378708755560900\) | \([4]\) | \(12582912\) | \(3.4728\) | |
| 46410.ck2 | 46410cn6 | \([1, 0, 0, -141270590, -644445412800]\) | \(311802066473807207098058600161/1033693082103011001480900\) | \(1033693082103011001480900\) | \([2, 2]\) | \(12582912\) | \(3.4728\) | |
| 46410.ck4 | 46410cn7 | \([1, 0, 0, -80405240, -1203420614130]\) | \(-57487943130312093140621093761/592356094985924086700006670\) | \(-592356094985924086700006670\) | \([2]\) | \(25165824\) | \(3.8194\) | |
| 46410.ck1 | 46410cn8 | \([1, 0, 0, -2258527940, -41313148043070]\) | \(1274090022584975661628188489514561/14072533302105480763470\) | \(14072533302105480763470\) | \([2]\) | \(25165824\) | \(3.8194\) |
Rank
sage: E.rank()
The elliptic curves in class 46410cn have rank \(0\).
Complex multiplication
The elliptic curves in class 46410cn do not have complex multiplication.Modular form 46410.2.a.cn
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
