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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 234135i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
234135.i2 | 234135i1 | \([1, -1, 1, -85081052, -302006562074]\) | \(1953326569433829507/262451171875\) | \(9151576611383056640625\) | \([2]\) | \(34406400\) | \(3.2329\) | \(\Gamma_0(N)\)-optimal |
234135.i1 | 234135i2 | \([1, -1, 1, -1361252927, -19330750155824]\) | \(8000051600110940079507/144453125\) | \(5037027766905234375\) | \([2]\) | \(68812800\) | \(3.5795\) |
Rank
sage: E.rank()
The elliptic curves in class 234135i have rank \(0\).
Complex multiplication
The elliptic curves in class 234135i do not have complex multiplication.Modular form 234135.2.a.i
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.