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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1392k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1392.e2 | 1392k1 | \([0, -1, 0, -16, -8768]\) | \(-117649/8118144\) | \(-33251917824\) | \([]\) | \(672\) | \(0.69784\) | \(\Gamma_0(N)\)-optimal |
1392.e1 | 1392k2 | \([0, -1, 0, -104176, 13014592]\) | \(-30526075007211889/103499257854\) | \(-423932960169984\) | \([]\) | \(4704\) | \(1.6708\) |
Rank
sage: E.rank()
The elliptic curves in class 1392k have rank \(1\).
Complex multiplication
The elliptic curves in class 1392k do not have complex multiplication.Modular form 1392.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.