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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 323400.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
323400.r1 | 323400r4 | \([0, -1, 0, -64680408, -200198341188]\) | \(15897679904620804/2475\) | \(4658900400000000\) | \([2]\) | \(18874368\) | \(2.8531\) | |
323400.r2 | 323400r5 | \([0, -1, 0, -34300408, 75850508812]\) | \(1185450336504002/26043266205\) | \(98046855224065440000000\) | \([2]\) | \(37748736\) | \(3.1996\) | |
323400.r3 | 323400r3 | \([0, -1, 0, -4655408, -2115841188]\) | \(5927735656804/2401490025\) | \(4520526399219600000000\) | \([2, 2]\) | \(18874368\) | \(2.8531\) | |
323400.r4 | 323400r2 | \([0, -1, 0, -4042908, -3126466188]\) | \(15529488955216/6125625\) | \(2882694622500000000\) | \([2, 2]\) | \(9437184\) | \(2.5065\) | |
323400.r5 | 323400r1 | \([0, -1, 0, -214783, -63966188]\) | \(-37256083456/38671875\) | \(-1137426855468750000\) | \([2]\) | \(4718592\) | \(2.1599\) | \(\Gamma_0(N)\)-optimal |
323400.r6 | 323400r6 | \([0, -1, 0, 15189592, -15411991188]\) | \(102949393183198/86815346805\) | \(-326839639560366240000000\) | \([2]\) | \(37748736\) | \(3.1996\) |
Rank
sage: E.rank()
The elliptic curves in class 323400.r have rank \(0\).
Complex multiplication
The elliptic curves in class 323400.r do not have complex multiplication.Modular form 323400.2.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.