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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 471510.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
471510.f1 | 471510f4 | \([1, -1, 0, -555984090, 4975856827956]\) | \(5401609226997647595049/86393158323264000\) | \(303995386843070421155904000\) | \([2]\) | \(306561024\) | \(3.8816\) | \(\Gamma_0(N)\)-optimal* |
471510.f2 | 471510f3 | \([1, -1, 0, -69264090, -102287620044]\) | \(10443846301537515049/4758933504000000\) | \(16745467576213868544000000\) | \([2]\) | \(153280512\) | \(3.5350\) | \(\Gamma_0(N)\)-optimal* |
471510.f3 | 471510f2 | \([1, -1, 0, -58594275, -169400210355]\) | \(6322686217296773689/135260510172840\) | \(475947076280357781651240\) | \([2]\) | \(102187008\) | \(3.3323\) | \(\Gamma_0(N)\)-optimal* |
471510.f4 | 471510f1 | \([1, -1, 0, -58290075, -171278523675]\) | \(6224721371657832889/2942222400\) | \(10352926713474446400\) | \([2]\) | \(51093504\) | \(2.9857\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 471510.f have rank \(0\).
Complex multiplication
The elliptic curves in class 471510.f do not have complex multiplication.Modular form 471510.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.