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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 15600z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 15600.n6 | 15600z1 | \([0, -1, 0, 5992, -73488]\) | \(371694959/249600\) | \(-15974400000000\) | \([2]\) | \(36864\) | \(1.2222\) | \(\Gamma_0(N)\)-optimal |
| 15600.n5 | 15600z2 | \([0, -1, 0, -26008, -585488]\) | \(30400540561/15210000\) | \(973440000000000\) | \([2, 2]\) | \(73728\) | \(1.5688\) | |
| 15600.n2 | 15600z3 | \([0, -1, 0, -338008, -75465488]\) | \(66730743078481/60937500\) | \(3900000000000000\) | \([2]\) | \(147456\) | \(1.9154\) | |
| 15600.n3 | 15600z4 | \([0, -1, 0, -226008, 41014512]\) | \(19948814692561/231344100\) | \(14806022400000000\) | \([2, 2]\) | \(147456\) | \(1.9154\) | |
| 15600.n1 | 15600z5 | \([0, -1, 0, -3606008, 2636854512]\) | \(81025909800741361/11088090\) | \(709637760000000\) | \([2]\) | \(294912\) | \(2.2619\) | |
| 15600.n4 | 15600z6 | \([0, -1, 0, -46008, 104374512]\) | \(-168288035761/73415764890\) | \(-4698608952960000000\) | \([2]\) | \(294912\) | \(2.2619\) |
Rank
sage: E.rank()
The elliptic curves in class 15600z have rank \(0\).
Complex multiplication
The elliptic curves in class 15600z do not have complex multiplication.Modular form 15600.2.a.z
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.
