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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 12342bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12342.bd2 | 12342bd1 | \([1, 0, 0, -30918, -2094300]\) | \(1845026709625/793152\) | \(1405117150272\) | \([2]\) | \(34560\) | \(1.2911\) | \(\Gamma_0(N)\)-optimal |
| 12342.bd3 | 12342bd2 | \([1, 0, 0, -26078, -2770932]\) | \(-1107111813625/1228691592\) | \(-2176702105415112\) | \([2]\) | \(69120\) | \(1.6376\) | |
| 12342.bd1 | 12342bd3 | \([1, 0, 0, -90813, 7956081]\) | \(46753267515625/11591221248\) | \(20534555505328128\) | \([2]\) | \(103680\) | \(1.8404\) | |
| 12342.bd4 | 12342bd4 | \([1, 0, 0, 218947, 50517105]\) | \(655215969476375/1001033261568\) | \(-1773391485896667648\) | \([2]\) | \(207360\) | \(2.1869\) |
Rank
sage: E.rank()
The elliptic curves in class 12342bd have rank \(1\).
Complex multiplication
The elliptic curves in class 12342bd do not have complex multiplication.Modular form 12342.2.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 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.
