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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 164730.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
164730.a1 | 164730cx3 | \([1, 1, 0, -4230070608, 105891762826512]\) | \(346795165011870675497264041/121778756846679600\) | \(2939443146120951265892400\) | \([2]\) | \(212336640\) | \(4.0500\) | |
164730.a2 | 164730cx2 | \([1, 1, 0, -265568608, 1638839533312]\) | \(85814444987865209552041/1585867720793760000\) | \(38278991535532116769440000\) | \([2, 2]\) | \(106168320\) | \(3.7034\) | |
164730.a3 | 164730cx1 | \([1, 1, 0, -34368608, -38886386688]\) | \(186001322269702352041/80595993600000000\) | \(1945391356643558400000000\) | \([2]\) | \(53084160\) | \(3.3568\) | \(\Gamma_0(N)\)-optimal |
164730.a4 | 164730cx4 | \([1, 1, 0, -266608, 4764999120112]\) | \(-86826493040041/406364619140547159600\) | \(-9808654033663677762599012400\) | \([2]\) | \(212336640\) | \(4.0500\) |
Rank
sage: E.rank()
The elliptic curves in class 164730.a have rank \(1\).
Complex multiplication
The elliptic curves in class 164730.a do not have complex multiplication.Modular form 164730.2.a.a
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.