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SageMath
E = EllipticCurve("es1")
E.isogeny_class()
Elliptic curves in class 47040es
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47040.b1 | 47040es1 | \([0, -1, 0, -167841, 26479041]\) | \(197723452/375\) | \(991730245632000\) | \([2]\) | \(430080\) | \(1.7673\) | \(\Gamma_0(N)\)-optimal |
47040.b2 | 47040es2 | \([0, -1, 0, -112961, 44029665]\) | \(-30138446/140625\) | \(-743797684224000000\) | \([2]\) | \(860160\) | \(2.1139\) |
Rank
sage: E.rank()
The elliptic curves in class 47040es have rank \(1\).
Complex multiplication
The elliptic curves in class 47040es do not have complex multiplication.Modular form 47040.2.a.es
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.