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SageMath
E = EllipticCurve("hw1")
E.isogeny_class()
Elliptic curves in class 369600.hw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
369600.hw1 | 369600hw2 | \([0, -1, 0, -916833, -16718463]\) | \(665863066004/384359283\) | \(49197988224000000000\) | \([2]\) | \(10321920\) | \(2.4681\) | |
369600.hw2 | 369600hw1 | \([0, -1, 0, -646833, -199508463]\) | \(935299949456/2750517\) | \(88016544000000000\) | \([2]\) | \(5160960\) | \(2.1215\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 369600.hw have rank \(0\).
Complex multiplication
The elliptic curves in class 369600.hw do not have complex multiplication.Modular form 369600.2.a.hw
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.