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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 208725.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 208725.q1 | 208725y5 | \([1, 1, 1, -281639663, 1819117796156]\) | \(89254274298475942657/17457\) | \(483220943390625\) | \([2]\) | \(15728640\) | \(3.1188\) | |
| 208725.q2 | 208725y3 | \([1, 1, 1, -17602538, 28418014406]\) | \(21790813729717297/304746849\) | \(8435588008770140625\) | \([2, 2]\) | \(7864320\) | \(2.7722\) | |
| 208725.q3 | 208725y6 | \([1, 1, 1, -17103413, 30106055156]\) | \(-19989223566735457/2584262514273\) | \(-71534041938249846140625\) | \([2]\) | \(15728640\) | \(3.1188\) | |
| 208725.q4 | 208725y4 | \([1, 1, 1, -4262288, -2939196094]\) | \(309368403125137/44372288367\) | \(1228253367995795109375\) | \([2]\) | \(7864320\) | \(2.7722\) | |
| 208725.q5 | 208725y2 | \([1, 1, 1, -1131413, 417101906]\) | \(5786435182177/627352209\) | \(17365511042628890625\) | \([2, 2]\) | \(3932160\) | \(2.4256\) | |
| 208725.q6 | 208725y1 | \([1, 1, 1, 93712, 32412656]\) | \(3288008303/18259263\) | \(-505428097180359375\) | \([2]\) | \(1966080\) | \(2.0791\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 208725.q have rank \(1\).
Complex multiplication
The elliptic curves in class 208725.q do not have complex multiplication.Modular form 208725.2.a.q
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
