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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 21175r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21175.u2 | 21175r1 | \([0, -1, 1, -4033, -108032]\) | \(-262144/35\) | \(-968822421875\) | \([]\) | \(21600\) | \(1.0325\) | \(\Gamma_0(N)\)-optimal |
| 21175.u3 | 21175r2 | \([0, -1, 1, 26217, 270093]\) | \(71991296/42875\) | \(-1186807466796875\) | \([]\) | \(64800\) | \(1.5818\) | |
| 21175.u1 | 21175r3 | \([0, -1, 1, -397283, 100957218]\) | \(-250523582464/13671875\) | \(-378446258544921875\) | \([]\) | \(194400\) | \(2.1311\) |
Rank
sage: E.rank()
The elliptic curves in class 21175r have rank \(1\).
Complex multiplication
The elliptic curves in class 21175r do not have complex multiplication.Modular form 21175.2.a.r
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
