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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 19320t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19320.l3 | 19320t1 | \([0, -1, 0, -6735, 214992]\) | \(2111937254864896/132040125\) | \(2112642000\) | \([2]\) | \(18432\) | \(0.84831\) | \(\Gamma_0(N)\)-optimal |
| 19320.l2 | 19320t2 | \([0, -1, 0, -7140, 188100]\) | \(157267580823376/32806265625\) | \(8398404000000\) | \([2, 2]\) | \(36864\) | \(1.1949\) | |
| 19320.l1 | 19320t3 | \([0, -1, 0, -36120, -2466468]\) | \(5089545532199524/353759765625\) | \(362250000000000\) | \([2]\) | \(73728\) | \(1.5415\) | |
| 19320.l4 | 19320t4 | \([0, -1, 0, 15360, 1115100]\) | \(391353415004156/755885521125\) | \(-774026773632000\) | \([4]\) | \(73728\) | \(1.5415\) |
Rank
sage: E.rank()
The elliptic curves in class 19320t have rank \(1\).
Complex multiplication
The elliptic curves in class 19320t do not have complex multiplication.Modular form 19320.2.a.t
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
