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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 126350q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 126350.g5 | 126350q1 | \([1, 0, 1, -4701, -251452]\) | \(-15625/28\) | \(-20582572937500\) | \([2]\) | \(342144\) | \(1.2449\) | \(\Gamma_0(N)\)-optimal |
| 126350.g4 | 126350q2 | \([1, 0, 1, -94951, -11261952]\) | \(128787625/98\) | \(72039005281250\) | \([2]\) | \(684288\) | \(1.5914\) | |
| 126350.g6 | 126350q3 | \([1, 0, 1, 40424, 5253798]\) | \(9938375/21952\) | \(-16136737183000000\) | \([2]\) | \(1026432\) | \(1.7942\) | |
| 126350.g3 | 126350q4 | \([1, 0, 1, -320576, 57237798]\) | \(4956477625/941192\) | \(691862606721125000\) | \([2]\) | \(2052864\) | \(2.1407\) | |
| 126350.g2 | 126350q5 | \([1, 0, 1, -1538951, 736820298]\) | \(-548347731625/1835008\) | \(-1348899500032000000\) | \([2]\) | \(3079296\) | \(2.3435\) | |
| 126350.g1 | 126350q6 | \([1, 0, 1, -24642951, 47083444298]\) | \(2251439055699625/25088\) | \(18441985352000000\) | \([2]\) | \(6158592\) | \(2.6900\) |
Rank
sage: E.rank()
The elliptic curves in class 126350q have rank \(0\).
Complex multiplication
The elliptic curves in class 126350q do not have complex multiplication.Modular form 126350.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
