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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 136367.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
136367.e1 | 136367e2 | \([1, -1, 1, -289149972, -1892414450440]\) | \(4399901392374538640127/64009\) | \(38894775880807\) | \([2]\) | \(19169280\) | \(3.0907\) | |
136367.e2 | 136367e1 | \([1, -1, 1, -18071857, -29565644160]\) | \(-1074191725926252207/4097152081\) | \(-2489615709354575263\) | \([2]\) | \(9584640\) | \(2.7441\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 136367.e have rank \(0\).
Complex multiplication
The elliptic curves in class 136367.e do not have complex multiplication.Modular form 136367.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.