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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 67600dd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67600.dg2 | 67600dd1 | \([0, -1, 0, -8823208, 12757644912]\) | \(-9836106385/3407872\) | \(-26318635584716800000000\) | \([]\) | \(7257600\) | \(3.0134\) | \(\Gamma_0(N)\)-optimal |
67600.dg1 | 67600dd2 | \([0, -1, 0, -765943208, 8159368844912]\) | \(-6434774386429585/140608\) | \(-1085900735795200000000\) | \([]\) | \(21772800\) | \(3.5627\) |
Rank
sage: E.rank()
The elliptic curves in class 67600dd have rank \(1\).
Complex multiplication
The elliptic curves in class 67600dd do not have complex multiplication.Modular form 67600.2.a.dd
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.