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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 41070.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 41070.bd1 | 41070be4 | \([1, 0, 0, -14357416, -9348082624]\) | \(127568139540190201/59114336463360\) | \(151671214214554412874240\) | \([2]\) | \(8273664\) | \(3.1426\) | |
| 41070.bd2 | 41070be2 | \([1, 0, 0, -7272841, 7548259121]\) | \(16581570075765001/998001000\) | \(2560597521908409000\) | \([2]\) | \(2757888\) | \(2.5933\) | |
| 41070.bd3 | 41070be1 | \([1, 0, 0, -427841, 132386121]\) | \(-3375675045001/999000000\) | \(-2563160682591000000\) | \([2]\) | \(1378944\) | \(2.2467\) | \(\Gamma_0(N)\)-optimal |
| 41070.bd4 | 41070be3 | \([1, 0, 0, 3165784, -1101664704]\) | \(1367594037332999/995878502400\) | \(-2555151773763049881600\) | \([2]\) | \(4136832\) | \(2.7960\) |
Rank
sage: E.rank()
The elliptic curves in class 41070.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 41070.bd do not have complex multiplication.Modular form 41070.2.a.bd
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
