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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 338130.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
338130.j1 | 338130j4 | \([1, -1, 0, -3110850, 2005178386]\) | \(189208196468929/10860320250\) | \(191101320730028270250\) | \([2]\) | \(11943936\) | \(2.6457\) | |
338130.j2 | 338130j2 | \([1, -1, 0, -535860, -150181880]\) | \(967068262369/4928040\) | \(86715210134840040\) | \([2]\) | \(3981312\) | \(2.0964\) | |
338130.j3 | 338130j1 | \([1, -1, 0, -15660, -4838000]\) | \(-24137569/561600\) | \(-9882075229041600\) | \([2]\) | \(1990656\) | \(1.7499\) | \(\Gamma_0(N)\)-optimal |
338130.j4 | 338130j3 | \([1, -1, 0, 140400, 127906636]\) | \(17394111071/411937500\) | \(-7248570806024437500\) | \([2]\) | \(5971968\) | \(2.2992\) |
Rank
sage: E.rank()
The elliptic curves in class 338130.j have rank \(1\).
Complex multiplication
The elliptic curves in class 338130.j do not have complex multiplication.Modular form 338130.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.