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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 416955t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
416955.t3 | 416955t1 | \([1, 0, 0, -73471, -7665760]\) | \(932288503609/779625\) | \(36678144974625\) | \([2]\) | \(1990656\) | \(1.5307\) | \(\Gamma_0(N)\)-optimal* |
416955.t2 | 416955t2 | \([1, 0, 0, -89716, -4030129]\) | \(1697509118089/833765625\) | \(39225238375640625\) | \([2, 2]\) | \(3981312\) | \(1.8772\) | \(\Gamma_0(N)\)-optimal* |
416955.t1 | 416955t3 | \([1, 0, 0, -766591, 255483746]\) | \(1058993490188089/13182390375\) | \(620177168877795375\) | \([2]\) | \(7962624\) | \(2.2238\) | \(\Gamma_0(N)\)-optimal* |
416955.t4 | 416955t4 | \([1, 0, 0, 327239, -30798640]\) | \(82375335041831/56396484375\) | \(-2653222292724609375\) | \([2]\) | \(7962624\) | \(2.2238\) |
Rank
sage: E.rank()
The elliptic curves in class 416955t have rank \(0\).
Complex multiplication
The elliptic curves in class 416955t do not have complex multiplication.Modular form 416955.2.a.t
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.