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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 47040ct
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47040.du2 | 47040ct1 | \([0, 1, 0, -11776, 486590]\) | \(69934528/225\) | \(581091940800\) | \([2]\) | \(114688\) | \(1.1235\) | \(\Gamma_0(N)\)-optimal |
47040.du1 | 47040ct2 | \([0, 1, 0, -16921, 14279]\) | \(3241792/1875\) | \(309915701760000\) | \([2]\) | \(229376\) | \(1.4701\) |
Rank
sage: E.rank()
The elliptic curves in class 47040ct have rank \(1\).
Complex multiplication
The elliptic curves in class 47040ct do not have complex multiplication.Modular form 47040.2.a.ct
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.