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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 425880i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
425880.i4 | 425880i1 | \([0, 0, 0, 13182, -173563]\) | \(4499456/2835\) | \(-159610216998960\) | \([2]\) | \(1179648\) | \(1.4143\) | \(\Gamma_0(N)\)-optimal* |
425880.i3 | 425880i2 | \([0, 0, 0, -55263, -1419262]\) | \(20720464/11025\) | \(9931302391046400\) | \([2, 2]\) | \(2359296\) | \(1.7609\) | \(\Gamma_0(N)\)-optimal* |
425880.i2 | 425880i3 | \([0, 0, 0, -511563, 139759958]\) | \(4108974916/36015\) | \(129769017909672960\) | \([2]\) | \(4718592\) | \(2.1075\) | \(\Gamma_0(N)\)-optimal* |
425880.i1 | 425880i4 | \([0, 0, 0, -694083, -222323218]\) | \(10262905636/13125\) | \(47291916147840000\) | \([2]\) | \(4718592\) | \(2.1075\) |
Rank
sage: E.rank()
The elliptic curves in class 425880i have rank \(1\).
Complex multiplication
The elliptic curves in class 425880i do not have complex multiplication.Modular form 425880.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.