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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 7007c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7007.b4 | 7007c1 | \([1, -1, 1, -769, 69360]\) | \(-426957777/17320303\) | \(-2037716327647\) | \([4]\) | \(7296\) | \(1.0420\) | \(\Gamma_0(N)\)-optimal |
| 7007.b3 | 7007c2 | \([1, -1, 1, -30414, 2037788]\) | \(26444947540257/169338169\) | \(19922466244681\) | \([2, 2]\) | \(14592\) | \(1.3886\) | |
| 7007.b2 | 7007c3 | \([1, -1, 1, -49279, -776870]\) | \(112489728522417/62811265517\) | \(7389682576809533\) | \([2]\) | \(29184\) | \(1.7352\) | |
| 7007.b1 | 7007c4 | \([1, -1, 1, -485869, 130476098]\) | \(107818231938348177/4463459\) | \(525121487891\) | \([2]\) | \(29184\) | \(1.7352\) |
Rank
sage: E.rank()
The elliptic curves in class 7007c have rank \(0\).
Complex multiplication
The elliptic curves in class 7007c do not have complex multiplication.Modular form 7007.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
