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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 12789c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12789.j2 | 12789c1 | \([0, 0, 1, -4116, -100242]\) | \(1835008/29\) | \(121873657941\) | \([]\) | \(14112\) | \(0.92765\) | \(\Gamma_0(N)\)-optimal |
12789.j1 | 12789c2 | \([0, 0, 1, -34986, 2471229]\) | \(1126924288/24389\) | \(102495746328381\) | \([3]\) | \(42336\) | \(1.4770\) |
Rank
sage: E.rank()
The elliptic curves in class 12789c have rank \(1\).
Complex multiplication
The elliptic curves in class 12789c do not have complex multiplication.Modular form 12789.2.a.c
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.