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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 458640.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
458640.r1 | 458640r4 | \([0, 0, 0, -55343883, -158471897542]\) | \(53365044437418169/41984670\) | \(14749131929047326720\) | \([2]\) | \(28311552\) | \(2.9847\) | |
458640.r2 | 458640r3 | \([0, 0, 0, -8068683, 5334694778]\) | \(165369706597369/60703354530\) | \(21324968958877196820480\) | \([2]\) | \(28311552\) | \(2.9847\) | \(\Gamma_0(N)\)-optimal* |
458640.r3 | 458640r2 | \([0, 0, 0, -3482283, -2441087782]\) | \(13293525831769/365192100\) | \(128291265891713433600\) | \([2, 2]\) | \(14155776\) | \(2.6382\) | \(\Gamma_0(N)\)-optimal* |
458640.r4 | 458640r1 | \([0, 0, 0, 45717, -124602982]\) | \(30080231/19110000\) | \(-6713305384181760000\) | \([2]\) | \(7077888\) | \(2.2916\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 458640.r have rank \(0\).
Complex multiplication
The elliptic curves in class 458640.r do not have complex multiplication.Modular form 458640.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.