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SageMath
E = EllipticCurve("en1")
E.isogeny_class()
Elliptic curves in class 446160.en
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
446160.en1 | 446160en3 | \([0, 1, 0, -17535496, 28255340660]\) | \(30161840495801041/2799263610\) | \(55343148179949527040\) | \([2]\) | \(33030144\) | \(2.8265\) | \(\Gamma_0(N)\)-optimal* |
446160.en2 | 446160en4 | \([0, 1, 0, -6449096, -5994713100]\) | \(1500376464746641/83599963590\) | \(1652822248062502133760\) | \([2]\) | \(33030144\) | \(2.8265\) | |
446160.en3 | 446160en2 | \([0, 1, 0, -1176296, 372720180]\) | \(9104453457841/2226896100\) | \(44027093555383910400\) | \([2, 2]\) | \(16515072\) | \(2.4799\) | \(\Gamma_0(N)\)-optimal* |
446160.en4 | 446160en1 | \([0, 1, 0, 175704, 36883380]\) | \(30342134159/47190000\) | \(-932975070044160000\) | \([2]\) | \(8257536\) | \(2.1334\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 446160.en have rank \(1\).
Complex multiplication
The elliptic curves in class 446160.en do not have complex multiplication.Modular form 446160.2.a.en
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.