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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 353925dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 353925.dd5 | 353925dd1 | \([1, -1, 0, -653967, 292156816]\) | \(-1532808577/938223\) | \(-18932577379360734375\) | \([2]\) | \(7864320\) | \(2.4009\) | \(\Gamma_0(N)\)-optimal |
| 353925.dd4 | 353925dd2 | \([1, -1, 0, -11680092, 15364869691]\) | \(8732907467857/1656369\) | \(33424179817883765625\) | \([2, 2]\) | \(15728640\) | \(2.7475\) | |
| 353925.dd1 | 353925dd3 | \([1, -1, 0, -186872967, 983305504066]\) | \(35765103905346817/1287\) | \(25970613689109375\) | \([2]\) | \(31457280\) | \(3.0941\) | |
| 353925.dd3 | 353925dd4 | \([1, -1, 0, -12905217, 11945545816]\) | \(11779205551777/3763454409\) | \(75943450343989458140625\) | \([2, 2]\) | \(31457280\) | \(3.0941\) | |
| 353925.dd6 | 353925dd5 | \([1, -1, 0, 36508158, 81371337691]\) | \(266679605718863/296110251723\) | \(-5975264146762211884171875\) | \([2]\) | \(62914560\) | \(3.4406\) | |
| 353925.dd2 | 353925dd6 | \([1, -1, 0, -81920592, -276331675559]\) | \(3013001140430737/108679952667\) | \(2193073089716665989421875\) | \([2]\) | \(62914560\) | \(3.4406\) |
Rank
sage: E.rank()
The elliptic curves in class 353925dd have rank \(0\).
Complex multiplication
The elliptic curves in class 353925dd do not have complex multiplication.Modular form 353925.2.a.dd
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.
