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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 25886.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25886.d1 | 25886a6 | \([1, 1, 1, -5048733, 4364278259]\) | \(2251439055699625/25088\) | \(158590356173312\) | \([2]\) | \(471744\) | \(2.2937\) | |
| 25886.d2 | 25886a5 | \([1, 1, 1, -315293, 68208115]\) | \(-548347731625/1835008\) | \(-11599751765819392\) | \([2]\) | \(235872\) | \(1.9471\) | |
| 25886.d3 | 25886a4 | \([1, 1, 1, -65678, 5282947]\) | \(4956477625/941192\) | \(5949616330814408\) | \([2]\) | \(157248\) | \(1.7444\) | |
| 25886.d4 | 25886a2 | \([1, 1, 1, -19453, -1051727]\) | \(128787625/98\) | \(619493578802\) | \([2]\) | \(52416\) | \(1.1951\) | |
| 25886.d5 | 25886a1 | \([1, 1, 1, -963, -23683]\) | \(-15625/28\) | \(-176998165372\) | \([2]\) | \(26208\) | \(0.84852\) | \(\Gamma_0(N)\)-optimal |
| 25886.d6 | 25886a3 | \([1, 1, 1, 8282, 490339]\) | \(9938375/21952\) | \(-138766561651648\) | \([2]\) | \(78624\) | \(1.3978\) |
Rank
sage: E.rank()
The elliptic curves in class 25886.d have rank \(1\).
Complex multiplication
The elliptic curves in class 25886.d do not have complex multiplication.Modular form 25886.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
