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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 10241e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10241.f1 | 10241e1 | \([0, -1, 1, -1339, -21617]\) | \(-2258403328/480491\) | \(-56529285659\) | \([]\) | \(6912\) | \(0.78454\) | \(\Gamma_0(N)\)-optimal |
10241.f2 | 10241e2 | \([0, -1, 1, 9441, 124452]\) | \(790939860992/517504691\) | \(-60883909391459\) | \([]\) | \(20736\) | \(1.3338\) |
Rank
sage: E.rank()
The elliptic curves in class 10241e have rank \(1\).
Complex multiplication
The elliptic curves in class 10241e do not have complex multiplication.Modular form 10241.2.a.e
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.