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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 25410.t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25410.t1 | 25410t6 | \([1, 1, 0, -5728747, 5274492601]\) | \(11736717412386894481/1890645330420\) | \(3349393532204185620\) | \([2]\) | \(983040\) | \(2.5642\) | |
| 25410.t2 | 25410t4 | \([1, 1, 0, -2386727, -1420221051]\) | \(848742840525560401/1443750000\) | \(2557691193750000\) | \([2]\) | \(491520\) | \(2.2176\) | |
| 25410.t3 | 25410t3 | \([1, 1, 0, -392647, 65391781]\) | \(3778993806976081/1138958528400\) | \(2017734509530832400\) | \([2, 2]\) | \(491520\) | \(2.2176\) | |
| 25410.t4 | 25410t2 | \([1, 1, 0, -150647, -21776619]\) | \(213429068128081/8537760000\) | \(15125162643360000\) | \([2, 2]\) | \(245760\) | \(1.8710\) | |
| 25410.t5 | 25410t1 | \([1, 1, 0, 4233, -1239531]\) | \(4733169839/378470400\) | \(-670483400294400\) | \([2]\) | \(122880\) | \(1.5244\) | \(\Gamma_0(N)\)-optimal |
| 25410.t6 | 25410t5 | \([1, 1, 0, 1071453, 439908561]\) | \(76786760064334319/91531319653620\) | \(-162153316176886700820\) | \([2]\) | \(983040\) | \(2.5642\) |
Rank
sage: E.rank()
The elliptic curves in class 25410.t have rank \(0\).
Complex multiplication
The elliptic curves in class 25410.t do not have complex multiplication.Modular form 25410.2.a.t
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
