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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 912j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
912.j2 | 912j1 | \([0, 1, 0, 3, -18]\) | \(131072/9747\) | \(-155952\) | \([2]\) | \(72\) | \(-0.32402\) | \(\Gamma_0(N)\)-optimal |
912.j1 | 912j2 | \([0, 1, 0, -92, -360]\) | \(340062928/13851\) | \(3545856\) | \([2]\) | \(144\) | \(0.022556\) |
Rank
sage: E.rank()
The elliptic curves in class 912j have rank \(0\).
Complex multiplication
The elliptic curves in class 912j do not have complex multiplication.Modular form 912.2.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 Cremona 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 Cremona labels.