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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 10010d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10010.d3 | 10010d1 | \([1, -1, 0, -8030, 278980]\) | \(57266517014673849/6166160\) | \(6166160\) | \([2]\) | \(7168\) | \(0.72990\) | \(\Gamma_0(N)\)-optimal |
| 10010.d2 | 10010d2 | \([1, -1, 0, -8050, 277536]\) | \(57695467871815929/594086392900\) | \(594086392900\) | \([2, 2]\) | \(14336\) | \(1.0765\) | |
| 10010.d1 | 10010d3 | \([1, -1, 0, -14420, -218050]\) | \(331616731345462809/170683258996250\) | \(170683258996250\) | \([2]\) | \(28672\) | \(1.4230\) | |
| 10010.d4 | 10010d4 | \([1, -1, 0, -2000, 680466]\) | \(-884984855328729/199224662446810\) | \(-199224662446810\) | \([2]\) | \(28672\) | \(1.4230\) |
Rank
sage: E.rank()
The elliptic curves in class 10010d have rank \(1\).
Complex multiplication
The elliptic curves in class 10010d do not have complex multiplication.Modular form 10010.2.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 Cremona 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 Cremona labels.
