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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 137904.x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 137904.x1 | 137904bo3 | \([0, -1, 0, -2029408, 841963264]\) | \(46753267515625/11591221248\) | \(229165510823270940672\) | \([2]\) | \(3981312\) | \(2.6170\) | |
| 137904.x2 | 137904bo1 | \([0, -1, 0, -690928, -220741184]\) | \(1845026709625/793152\) | \(15681098596220928\) | \([2]\) | \(1327104\) | \(2.0677\) | \(\Gamma_0(N)\)-optimal |
| 137904.x3 | 137904bo2 | \([0, -1, 0, -582768, -292299840]\) | \(-1107111813625/1228691592\) | \(-24291981862870745088\) | \([2]\) | \(2654208\) | \(2.4143\) | |
| 137904.x4 | 137904bo4 | \([0, -1, 0, 4892832, 5333112576]\) | \(655215969476375/1001033261568\) | \(-19791037875141740593152\) | \([2]\) | \(7962624\) | \(2.9636\) |
Rank
sage: E.rank()
The elliptic curves in class 137904.x have rank \(0\).
Complex multiplication
The elliptic curves in class 137904.x do not have complex multiplication.Modular form 137904.2.a.x
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.
