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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 12705.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12705.d1 | 12705k4 | \([1, 0, 0, -248476, 47652551]\) | \(957681397954009/31185\) | \(55246129785\) | \([2]\) | \(46080\) | \(1.5625\) | |
| 12705.d2 | 12705k3 | \([1, 0, 0, -24626, -225939]\) | \(932288503609/527295615\) | \(934136347005015\) | \([2]\) | \(46080\) | \(1.5625\) | |
| 12705.d3 | 12705k2 | \([1, 0, 0, -15551, 741456]\) | \(234770924809/1334025\) | \(2363306663025\) | \([2, 2]\) | \(23040\) | \(1.2160\) | |
| 12705.d4 | 12705k1 | \([1, 0, 0, -426, 24531]\) | \(-4826809/144375\) | \(-255769119375\) | \([2]\) | \(11520\) | \(0.86939\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12705.d have rank \(0\).
Complex multiplication
The elliptic curves in class 12705.d do not have complex multiplication.Modular form 12705.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
