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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 194810.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
194810.c1 | 194810p4 | \([1, 0, 1, -20186554, 34907522802]\) | \(513516182162686336369/1944885031250\) | \(3445482470846281250\) | \([2]\) | \(16174080\) | \(2.7716\) | |
194810.c2 | 194810p3 | \([1, 0, 1, -1280304, 528397802]\) | \(131010595463836369/7704101562500\) | \(13648285868164062500\) | \([2]\) | \(8087040\) | \(2.4250\) | |
194810.c3 | 194810p2 | \([1, 0, 1, -343764, 8279786]\) | \(2535986675931409/1450751712200\) | \(2570095154016744200\) | \([2]\) | \(5391360\) | \(2.2223\) | |
194810.c4 | 194810p1 | \([1, 0, 1, -222764, -40313814]\) | \(690080604747409/3406760000\) | \(6035283152360000\) | \([2]\) | \(2695680\) | \(1.8757\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 194810.c have rank \(1\).
Complex multiplication
The elliptic curves in class 194810.c do not have complex multiplication.Modular form 194810.2.a.c
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.