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SageMath
E = EllipticCurve("ij1")
E.isogeny_class()
Elliptic curves in class 122304ij
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 122304.hn4 | 122304ij1 | \([0, 1, 0, -61217, 121692255]\) | \(-822656953/207028224\) | \(-6384953203196166144\) | \([2]\) | \(2211840\) | \(2.2876\) | \(\Gamma_0(N)\)-optimal |
| 122304.hn3 | 122304ij2 | \([0, 1, 0, -4075297, 3136266335]\) | \(242702053576633/2554695936\) | \(78789324878502690816\) | \([2, 2]\) | \(4423680\) | \(2.6342\) | |
| 122304.hn2 | 122304ij3 | \([0, 1, 0, -7336737, -2596692897]\) | \(1416134368422073/725251155408\) | \(22367455984378391298048\) | \([2]\) | \(8847360\) | \(2.9808\) | |
| 122304.hn1 | 122304ij4 | \([0, 1, 0, -65039137, 201866191967]\) | \(986551739719628473/111045168\) | \(3424741744976068608\) | \([2]\) | \(8847360\) | \(2.9808\) |
Rank
sage: E.rank()
The elliptic curves in class 122304ij have rank \(1\).
Complex multiplication
The elliptic curves in class 122304ij do not have complex multiplication.Modular form 122304.2.a.ij
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.
