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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 68970.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68970.e1 | 68970e4 | \([1, 1, 0, -75143, 7885563]\) | \(26487576322129/44531250\) | \(78889825781250\) | \([2]\) | \(327680\) | \(1.5619\) | |
68970.e2 | 68970e2 | \([1, 1, 0, -6173, 36777]\) | \(14688124849/8122500\) | \(14389504222500\) | \([2, 2]\) | \(163840\) | \(1.2153\) | |
68970.e3 | 68970e1 | \([1, 1, 0, -3753, -89547]\) | \(3301293169/22800\) | \(40391590800\) | \([2]\) | \(81920\) | \(0.86872\) | \(\Gamma_0(N)\)-optimal |
68970.e4 | 68970e3 | \([1, 1, 0, 24077, 321127]\) | \(871257511151/527800050\) | \(-935029984378050\) | \([2]\) | \(327680\) | \(1.5619\) |
Rank
sage: E.rank()
The elliptic curves in class 68970.e have rank \(1\).
Complex multiplication
The elliptic curves in class 68970.e do not have complex multiplication.Modular form 68970.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.