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SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 121680dl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121680.ca3 | 121680dl1 | \([0, 0, 0, -146523, 138573578]\) | \(-24137569/561600\) | \(-8094214128343449600\) | \([2]\) | \(1548288\) | \(2.3089\) | \(\Gamma_0(N)\)-optimal |
121680.ca2 | 121680dl2 | \([0, 0, 0, -5013723, 4301976458]\) | \(967068262369/4928040\) | \(71026728976213770240\) | \([2]\) | \(3096576\) | \(2.6554\) | |
121680.ca4 | 121680dl3 | \([0, 0, 0, 1313637, -3661638838]\) | \(17394111071/411937500\) | \(-5937162272960256000000\) | \([2]\) | \(4644864\) | \(2.8582\) | |
121680.ca1 | 121680dl4 | \([0, 0, 0, -29106363, -57365106838]\) | \(189208196468929/10860320250\) | \(156527346164324189184000\) | \([2]\) | \(9289728\) | \(3.2048\) |
Rank
sage: E.rank()
The elliptic curves in class 121680dl have rank \(1\).
Complex multiplication
The elliptic curves in class 121680dl do not have complex multiplication.Modular form 121680.2.a.dl
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.