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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 36414p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
36414.w1 | 36414p1 | \([1, -1, 0, -1975947, -1009423675]\) | \(9869198625/614656\) | \(53137356615966083328\) | \([2]\) | \(1114112\) | \(2.5364\) | \(\Gamma_0(N)\)-optimal |
36414.w2 | 36414p2 | \([1, -1, 0, 1561413, -4230543691]\) | \(4869777375/92236816\) | \(-7973924577183410379408\) | \([2]\) | \(2228224\) | \(2.8829\) |
Rank
sage: E.rank()
The elliptic curves in class 36414p have rank \(0\).
Complex multiplication
The elliptic curves in class 36414p do not have complex multiplication.Modular form 36414.2.a.p
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.