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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 8015.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8015.d1 | 8015e3 | \([1, -1, 0, -2399, -21452]\) | \(1527271621918281/673764327845\) | \(673764327845\) | \([2]\) | \(9408\) | \(0.96545\) | |
| 8015.d2 | 8015e2 | \([1, -1, 0, -1174, 15543]\) | \(179034228973881/3147771025\) | \(3147771025\) | \([2, 2]\) | \(4704\) | \(0.61887\) | |
| 8015.d3 | 8015e1 | \([1, -1, 0, -1169, 15680]\) | \(176756829459561/56105\) | \(56105\) | \([2]\) | \(2352\) | \(0.27230\) | \(\Gamma_0(N)\)-optimal |
| 8015.d4 | 8015e4 | \([1, -1, 0, -29, 43710]\) | \(-2749884201/825087143125\) | \(-825087143125\) | \([4]\) | \(9408\) | \(0.96545\) |
Rank
sage: E.rank()
The elliptic curves in class 8015.d have rank \(0\).
Complex multiplication
The elliptic curves in class 8015.d do not have complex multiplication.Modular form 8015.2.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.
