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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 137904be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
137904.i2 | 137904be1 | \([0, -1, 0, 1602896, -1710533696]\) | \(50611530622079699/169662750916608\) | \(-1526780165176474730496\) | \([2]\) | \(5419008\) | \(2.7479\) | \(\Gamma_0(N)\)-optimal |
137904.i1 | 137904be2 | \([0, -1, 0, -15436464, -20208462912]\) | \(45204035637810785581/6545053349462016\) | \(58898359127113929326592\) | \([2]\) | \(10838016\) | \(3.0944\) |
Rank
sage: E.rank()
The elliptic curves in class 137904be have rank \(0\).
Complex multiplication
The elliptic curves in class 137904be do not have complex multiplication.Modular form 137904.2.a.be
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.