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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 176001.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 176001.d1 | 176001d3 | \([1, 0, 0, -59136342, 175032046917]\) | \(947531277805646290177/38367\) | \(926086109823\) | \([2]\) | \(7864320\) | \(2.7073\) | |
| 176001.d2 | 176001d6 | \([1, 0, 0, -12273547, -13455000118]\) | \(8471112631466271697/1662662681263647\) | \(40132635192726286654143\) | \([2]\) | \(15728640\) | \(3.0538\) | |
| 176001.d3 | 176001d4 | \([1, 0, 0, -3766832, 2624392575]\) | \(244883173420511137/18418027974129\) | \(444566421069468952401\) | \([2, 2]\) | \(7864320\) | \(2.7073\) | |
| 176001.d4 | 176001d2 | \([1, 0, 0, -3696027, 2734635960]\) | \(231331938231569617/1472026689\) | \(35531145775579041\) | \([2, 2]\) | \(3932160\) | \(2.3607\) | |
| 176001.d5 | 176001d1 | \([1, 0, 0, -226582, 44428307]\) | \(-53297461115137/4513839183\) | \(-108953104734566127\) | \([2]\) | \(1966080\) | \(2.0141\) | \(\Gamma_0(N)\)-optimal |
| 176001.d6 | 176001d5 | \([1, 0, 0, 3607003, 11648491848]\) | \(215015459663151503/2552757445339983\) | \(-61617358977157568121327\) | \([2]\) | \(15728640\) | \(3.0538\) |
Rank
sage: E.rank()
The elliptic curves in class 176001.d have rank \(0\).
Complex multiplication
The elliptic curves in class 176001.d do not have complex multiplication.Modular form 176001.2.a.d
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
