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SageMath
E = EllipticCurve("dh1")
E.isogeny_class()
Elliptic curves in class 121680dh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121680.v2 | 121680dh1 | \([0, 0, 0, 28392, 518492]\) | \(2809856/1755\) | \(-1580901196942080\) | \([]\) | \(387072\) | \(1.6055\) | \(\Gamma_0(N)\)-optimal |
121680.v1 | 121680dh2 | \([0, 0, 0, -336648, -85849972]\) | \(-4684079104/823875\) | \(-742145284120032000\) | \([]\) | \(1161216\) | \(2.1548\) |
Rank
sage: E.rank()
The elliptic curves in class 121680dh have rank \(1\).
Complex multiplication
The elliptic curves in class 121680dh do not have complex multiplication.Modular form 121680.2.a.dh
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.