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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 229320be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 229320.q4 | 229320be1 | \([0, 0, 0, 4290342, 3717725893]\) | \(6364491337435136/8034291412875\) | \(-11025120151454371326000\) | \([2]\) | \(17694720\) | \(2.9156\) | \(\Gamma_0(N)\)-optimal |
| 229320.q3 | 229320be2 | \([0, 0, 0, -25893903, 35960536402]\) | \(87450143958975184/25164018140625\) | \(552504377521929924000000\) | \([2, 2]\) | \(35389440\) | \(3.2621\) | |
| 229320.q1 | 229320be3 | \([0, 0, 0, -379849323, 2849127423478]\) | \(69014771940559650916/9797607421875\) | \(860470050462750000000000\) | \([2]\) | \(70778880\) | \(3.6087\) | |
| 229320.q2 | 229320be4 | \([0, 0, 0, -154886403, -713666478098]\) | \(4678944235881273796/202428825314625\) | \(17778211968841718599296000\) | \([2]\) | \(70778880\) | \(3.6087\) |
Rank
sage: E.rank()
The elliptic curves in class 229320be have rank \(1\).
Complex multiplication
The elliptic curves in class 229320be do not have complex multiplication.Modular form 229320.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
