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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 328560df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 328560.df4 | 328560df1 | \([0, 1, 0, -31177902160, 1992587693550740]\) | \(318929057401476905525449/21353131537921474560\) | \(224404658182946456562948731043840\) | \([2]\) | \(1617960960\) | \(4.9566\) | \(\Gamma_0(N)\)-optimal |
| 328560.df2 | 328560df2 | \([0, 1, 0, -490538076240, 132237550267427988]\) | \(1242142983306846366056931529/6179359141291622400\) | \(64940216072222629732908898713600\) | \([2, 2]\) | \(3235921920\) | \(5.3032\) | |
| 328560.df1 | 328560df3 | \([0, 1, 0, -7848599849040, 8463232285487099028]\) | \(5087799435928552778197163696329/125914832087040\) | \(1323266088839246293562818560\) | \([2]\) | \(6471843840\) | \(5.6498\) | |
| 328560.df3 | 328560df4 | \([0, 1, 0, -482239088720, 136927769538686100]\) | \(-1180159344892952613848670409/87759036144023189760000\) | \(-922278611612081461148263923056640000\) | \([4]\) | \(6471843840\) | \(5.6498\) |
Rank
sage: E.rank()
The elliptic curves in class 328560df have rank \(0\).
Complex multiplication
The elliptic curves in class 328560df do not have complex multiplication.Modular form 328560.2.a.df
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
