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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 57222.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 57222.bm1 | 57222bh3 | \([1, -1, 1, -209435, 36802059]\) | \(57736239625/255552\) | \(4496766540121152\) | \([2]\) | \(442368\) | \(1.8558\) | |
| 57222.bm2 | 57222bh4 | \([1, -1, 1, -105395, 73299291]\) | \(-7357983625/127552392\) | \(-2244448599337969992\) | \([2]\) | \(884736\) | \(2.2024\) | |
| 57222.bm3 | 57222bh1 | \([1, -1, 1, -14360, -621129]\) | \(18609625/1188\) | \(20904389907588\) | \([2]\) | \(147456\) | \(1.3065\) | \(\Gamma_0(N)\)-optimal |
| 57222.bm4 | 57222bh2 | \([1, -1, 1, 11650, -2639505]\) | \(9938375/176418\) | \(-3104301901276818\) | \([2]\) | \(294912\) | \(1.6531\) |
Rank
sage: E.rank()
The elliptic curves in class 57222.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 57222.bm do not have complex multiplication.Modular form 57222.2.a.bm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
