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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1040.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1040.d1 | 1040a3 | \([0, 0, 0, -563, 5138]\) | \(9636491538/8125\) | \(16640000\) | \([2]\) | \(256\) | \(0.31312\) | |
| 1040.d2 | 1040a2 | \([0, 0, 0, -43, 42]\) | \(8586756/4225\) | \(4326400\) | \([2, 2]\) | \(128\) | \(-0.033453\) | |
| 1040.d3 | 1040a1 | \([0, 0, 0, -23, -42]\) | \(5256144/65\) | \(16640\) | \([2]\) | \(64\) | \(-0.38003\) | \(\Gamma_0(N)\)-optimal |
| 1040.d4 | 1040a4 | \([0, 0, 0, 157, 322]\) | \(208974222/142805\) | \(-292464640\) | \([2]\) | \(256\) | \(0.31312\) |
Rank
sage: E.rank()
The elliptic curves in class 1040.d have rank \(1\).
Complex multiplication
The elliptic curves in class 1040.d do not have complex multiplication.Modular form 1040.2.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.
