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SageMath
E = EllipticCurve("mr1")
E.isogeny_class()
Elliptic curves in class 141120mr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 141120.gq4 | 141120mr1 | \([0, 0, 0, -1297128, 568619912]\) | \(2748251600896/2205\) | \(193653039928320\) | \([2]\) | \(1572864\) | \(2.0469\) | \(\Gamma_0(N)\)-optimal |
| 141120.gq3 | 141120mr2 | \([0, 0, 0, -1305948, 560494928]\) | \(175293437776/4862025\) | \(6832079248671129600\) | \([2, 2]\) | \(3145728\) | \(2.3934\) | |
| 141120.gq5 | 141120mr3 | \([0, 0, 0, 281652, 1836290288]\) | \(439608956/259416045\) | \(-1458117535649722859520\) | \([2]\) | \(6291456\) | \(2.7400\) | |
| 141120.gq2 | 141120mr4 | \([0, 0, 0, -3034668, -1235299408]\) | \(549871953124/200930625\) | \(1129384528861962240000\) | \([2, 2]\) | \(6291456\) | \(2.7400\) | |
| 141120.gq6 | 141120mr5 | \([0, 0, 0, 9313332, -8737944208]\) | \(7947184069438/7533176175\) | \(-84684478787009534361600\) | \([2]\) | \(12582912\) | \(3.0866\) | |
| 141120.gq1 | 141120mr6 | \([0, 0, 0, -43042188, -108663492112]\) | \(784478485879202/221484375\) | \(2489824799078400000000\) | \([2]\) | \(12582912\) | \(3.0866\) |
Rank
sage: E.rank()
The elliptic curves in class 141120mr have rank \(1\).
Complex multiplication
The elliptic curves in class 141120mr do not have complex multiplication.Modular form 141120.2.a.mr
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
