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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 92778.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
92778.r1 | 92778m4 | \([1, 1, 1, -2968942, -1970259601]\) | \(268498407453697/252\) | \(2716362262908\) | \([2]\) | \(1695744\) | \(2.1142\) | |
92778.r2 | 92778m6 | \([1, 1, 1, -2019072, 1092828159]\) | \(84448510979617/933897762\) | \(10066685071869193698\) | \([2]\) | \(3391488\) | \(2.4607\) | |
92778.r3 | 92778m3 | \([1, 1, 1, -229782, -15100209]\) | \(124475734657/63011844\) | \(679218234753356676\) | \([2, 2]\) | \(1695744\) | \(2.1142\) | |
92778.r4 | 92778m2 | \([1, 1, 1, -185602, -30828289]\) | \(65597103937/63504\) | \(684523290252816\) | \([2, 2]\) | \(847872\) | \(1.7676\) | |
92778.r5 | 92778m1 | \([1, 1, 1, -8882, -715201]\) | \(-7189057/16128\) | \(-173847184826112\) | \([2]\) | \(423936\) | \(1.4210\) | \(\Gamma_0(N)\)-optimal |
92778.r6 | 92778m5 | \([1, 1, 1, 852628, -115547857]\) | \(6359387729183/4218578658\) | \(-45472967736905848482\) | \([2]\) | \(3391488\) | \(2.4607\) |
Rank
sage: E.rank()
The elliptic curves in class 92778.r have rank \(1\).
Complex multiplication
The elliptic curves in class 92778.r do not have complex multiplication.Modular form 92778.2.a.r
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.