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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 210786.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
210786.e1 | 210786be3 | \([1, 1, 0, -791410, 270656512]\) | \(8671983378625/82308\) | \(520298749837092\) | \([2]\) | \(2903040\) | \(1.9861\) | |
210786.e2 | 210786be4 | \([1, 1, 0, -772920, 283928634]\) | \(-8078253774625/846825858\) | \(-5353093687698921042\) | \([2]\) | \(5806080\) | \(2.3327\) | |
210786.e3 | 210786be1 | \([1, 1, 0, -14830, -59276]\) | \(57066625/32832\) | \(207542991624768\) | \([2]\) | \(967680\) | \(1.4368\) | \(\Gamma_0(N)\)-optimal |
210786.e4 | 210786be2 | \([1, 1, 0, 59130, -399492]\) | \(3616805375/2105352\) | \(-13308694337938248\) | \([2]\) | \(1935360\) | \(1.7834\) |
Rank
sage: E.rank()
The elliptic curves in class 210786.e have rank \(0\).
Complex multiplication
The elliptic curves in class 210786.e do not have complex multiplication.Modular form 210786.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.