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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 3825.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3825.d1 | 3825i3 | \([1, -1, 1, -20405, 1126972]\) | \(82483294977/17\) | \(193640625\) | \([2]\) | \(4096\) | \(0.97739\) | |
| 3825.d2 | 3825i2 | \([1, -1, 1, -1280, 17722]\) | \(20346417/289\) | \(3291890625\) | \([2, 2]\) | \(2048\) | \(0.63082\) | |
| 3825.d3 | 3825i1 | \([1, -1, 1, -155, -278]\) | \(35937/17\) | \(193640625\) | \([2]\) | \(1024\) | \(0.28424\) | \(\Gamma_0(N)\)-optimal |
| 3825.d4 | 3825i4 | \([1, -1, 1, -155, 46972]\) | \(-35937/83521\) | \(-951356390625\) | \([2]\) | \(4096\) | \(0.97739\) |
Rank
sage: E.rank()
The elliptic curves in class 3825.d have rank \(1\).
Complex multiplication
The elliptic curves in class 3825.d do not have complex multiplication.Modular form 3825.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
