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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 136242i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
136242.z1 | 136242i1 | \([1, -1, 1, -5204, 159067]\) | \(-35937/4\) | \(-1734504804036\) | \([]\) | \(275184\) | \(1.0858\) | \(\Gamma_0(N)\)-optimal |
136242.z2 | 136242i2 | \([1, -1, 1, 32641, -234521]\) | \(109503/64\) | \(-2247918226030656\) | \([]\) | \(825552\) | \(1.6351\) |
Rank
sage: E.rank()
The elliptic curves in class 136242i have rank \(0\).
Complex multiplication
The elliptic curves in class 136242i do not have complex multiplication.Modular form 136242.2.a.i
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.