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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 175329.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 175329.d1 | 175329e3 | \([1, -1, 1, -134696, 19056700]\) | \(209267191953/55223\) | \(71318735652087\) | \([2]\) | \(819200\) | \(1.6426\) | |
| 175329.d2 | 175329e2 | \([1, -1, 1, -9461, 221356]\) | \(72511713/25921\) | \(33476141224449\) | \([2, 2]\) | \(409600\) | \(1.2960\) | |
| 175329.d3 | 175329e1 | \([1, -1, 1, -4016, -94454]\) | \(5545233/161\) | \(207926343009\) | \([2]\) | \(204800\) | \(0.94947\) | \(\Gamma_0(N)\)-optimal |
| 175329.d4 | 175329e4 | \([1, -1, 1, 28654, 1532512]\) | \(2014698447/1958887\) | \(-2529839815390503\) | \([2]\) | \(819200\) | \(1.6426\) |
Rank
sage: E.rank()
The elliptic curves in class 175329.d have rank \(2\).
Complex multiplication
The elliptic curves in class 175329.d do not have complex multiplication.Modular form 175329.2.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.
