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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 304920r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 304920.r4 | 304920r1 | \([0, 0, 0, -800778, 275813813]\) | \(2748251600896/2205\) | \(45562989946320\) | \([2]\) | \(2621440\) | \(1.9263\) | \(\Gamma_0(N)\)-optimal |
| 304920.r3 | 304920r2 | \([0, 0, 0, -806223, 271872722]\) | \(175293437776/4862025\) | \(1607462285306169600\) | \([2, 2]\) | \(5242880\) | \(2.2729\) | |
| 304920.r5 | 304920r3 | \([0, 0, 0, 173877, 890707862]\) | \(439608956/259416045\) | \(-343068173068454507520\) | \([2]\) | \(10485760\) | \(2.6194\) | |
| 304920.r2 | 304920r4 | \([0, 0, 0, -1873443, -599192242]\) | \(549871953124/200930625\) | \(265723357366938240000\) | \([2, 2]\) | \(10485760\) | \(2.6194\) | |
| 304920.r6 | 304920r5 | \([0, 0, 0, 5749557, -4238412442]\) | \(7947184069438/7533176175\) | \(-19924696544965506201600\) | \([2]\) | \(20971520\) | \(2.9660\) | |
| 304920.r1 | 304920r6 | \([0, 0, 0, -26571963, -52708129738]\) | \(784478485879202/221484375\) | \(585809870738400000000\) | \([2]\) | \(20971520\) | \(2.9660\) |
Rank
sage: E.rank()
The elliptic curves in class 304920r have rank \(2\).
Complex multiplication
The elliptic curves in class 304920r do not have complex multiplication.Modular form 304920.2.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 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.
