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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 47040dz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47040.s2 | 47040dz1 | \([0, -1, 0, 19, -69]\) | \(229376/675\) | \(-2116800\) | \([]\) | \(6912\) | \(-0.10282\) | \(\Gamma_0(N)\)-optimal |
47040.s1 | 47040dz2 | \([0, -1, 0, -821, -8805]\) | \(-19539165184/46875\) | \(-147000000\) | \([]\) | \(20736\) | \(0.44649\) |
Rank
sage: E.rank()
The elliptic curves in class 47040dz have rank \(1\).
Complex multiplication
The elliptic curves in class 47040dz do not have complex multiplication.Modular form 47040.2.a.dz
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.