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SageMath
E = EllipticCurve("fr1")
E.isogeny_class()
Elliptic curves in class 54450fr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54450.gf3 | 54450fr1 | \([1, -1, 1, -150305, 21193197]\) | \(18609625/1188\) | \(23972874174562500\) | \([2]\) | \(552960\) | \(1.8936\) | \(\Gamma_0(N)\)-optimal |
| 54450.gf4 | 54450fr2 | \([1, -1, 1, 121945, 89255697]\) | \(9938375/176418\) | \(-3559971814922531250\) | \([2]\) | \(1105920\) | \(2.2402\) | |
| 54450.gf1 | 54450fr3 | \([1, -1, 1, -2192180, -1243952553]\) | \(57736239625/255552\) | \(5156831600217000000\) | \([2]\) | \(1658880\) | \(2.4429\) | |
| 54450.gf2 | 54450fr4 | \([1, -1, 1, -1103180, -2481056553]\) | \(-7357983625/127552392\) | \(-2573903572458310125000\) | \([2]\) | \(3317760\) | \(2.7895\) |
Rank
sage: E.rank()
The elliptic curves in class 54450fr have rank \(0\).
Complex multiplication
The elliptic curves in class 54450fr do not have complex multiplication.Modular form 54450.2.a.fr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
