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SageMath
E = EllipticCurve("br1")
E.isogeny_class()
Elliptic curves in class 123210br
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123210.z3 | 123210br1 | \([1, -1, 0, -3850569, -3574425267]\) | \(-3375675045001/999000000\) | \(-1868544137608839000000\) | \([2]\) | \(11031552\) | \(2.7960\) | \(\Gamma_0(N)\)-optimal |
| 123210.z2 | 123210br2 | \([1, -1, 0, -65455569, -203802996267]\) | \(16581570075765001/998001000\) | \(1866675593471230161000\) | \([2]\) | \(22063104\) | \(3.1426\) | |
| 123210.z4 | 123210br3 | \([1, -1, 0, 28492056, 29744947008]\) | \(1367594037332999/995878502400\) | \(-1862705643073263363686400\) | \([2]\) | \(33094656\) | \(3.3453\) | |
| 123210.z1 | 123210br4 | \([1, -1, 0, -129216744, 252398230848]\) | \(127568139540190201/59114336463360\) | \(110568315162410166985320960\) | \([2]\) | \(66189312\) | \(3.6919\) |
Rank
sage: E.rank()
The elliptic curves in class 123210br have rank \(1\).
Complex multiplication
The elliptic curves in class 123210br do not have complex multiplication.Modular form 123210.2.a.br
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
