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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 7098.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7098.f1 | 7098b4 | \([1, 1, 0, -227139, 41571873]\) | \(268498407453697/252\) | \(1216355868\) | \([2]\) | \(30720\) | \(1.4716\) | |
| 7098.f2 | 7098b5 | \([1, 1, 0, -154469, -23207517]\) | \(84448510979617/933897762\) | \(4507746122701458\) | \([2]\) | \(61440\) | \(1.8181\) | |
| 7098.f3 | 7098b3 | \([1, 1, 0, -17579, 310185]\) | \(124475734657/63011844\) | \(304146135725796\) | \([2, 2]\) | \(30720\) | \(1.4716\) | |
| 7098.f4 | 7098b2 | \([1, 1, 0, -14199, 644805]\) | \(65597103937/63504\) | \(306521678736\) | \([2, 2]\) | \(15360\) | \(1.1250\) | |
| 7098.f5 | 7098b1 | \([1, 1, 0, -679, 14773]\) | \(-7189057/16128\) | \(-77846775552\) | \([2]\) | \(7680\) | \(0.77843\) | \(\Gamma_0(N)\)-optimal |
| 7098.f6 | 7098b6 | \([1, 1, 0, 65231, 2479807]\) | \(6359387729183/4218578658\) | \(-20362273433642322\) | \([2]\) | \(61440\) | \(1.8181\) |
Rank
sage: E.rank()
The elliptic curves in class 7098.f have rank \(0\).
Complex multiplication
The elliptic curves in class 7098.f do not have complex multiplication.Modular form 7098.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
