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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 29645e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
29645.l2 | 29645e1 | \([1, 1, 0, -123, 3196208]\) | \(-1/21175\) | \(-4413343898384575\) | \([2]\) | \(184320\) | \(1.6808\) | \(\Gamma_0(N)\)-optimal |
29645.l1 | 29645e2 | \([1, 1, 0, -326218, 70436997]\) | \(18420660721/336875\) | \(70212289292481875\) | \([2]\) | \(368640\) | \(2.0274\) |
Rank
sage: E.rank()
The elliptic curves in class 29645e have rank \(1\).
Complex multiplication
The elliptic curves in class 29645e do not have complex multiplication.Modular form 29645.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.