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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 303450t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 303450.t5 | 303450t1 | \([1, 1, 0, -29050, 4196500]\) | \(-7189057/16128\) | \(-6082667388000000\) | \([2]\) | \(2621440\) | \(1.7173\) | \(\Gamma_0(N)\)-optimal |
| 303450.t4 | 303450t2 | \([1, 1, 0, -607050, 181642500]\) | \(65597103937/63504\) | \(23950502840250000\) | \([2, 2]\) | \(5242880\) | \(2.0639\) | |
| 303450.t1 | 303450t3 | \([1, 1, 0, -9710550, 11642949000]\) | \(268498407453697/252\) | \(95041677937500\) | \([2]\) | \(10485760\) | \(2.4104\) | |
| 303450.t3 | 303450t4 | \([1, 1, 0, -751550, 88440000]\) | \(124475734657/63011844\) | \(23764886443238062500\) | \([2, 2]\) | \(10485760\) | \(2.4104\) | |
| 303450.t6 | 303450t5 | \([1, 1, 0, 2788700, 686742250]\) | \(6359387729183/4218578658\) | \(-1591034897490662531250\) | \([2]\) | \(20971520\) | \(2.7570\) | |
| 303450.t2 | 303450t6 | \([1, 1, 0, -6603800, -6471932250]\) | \(84448510979617/933897762\) | \(352219088581571531250\) | \([2]\) | \(20971520\) | \(2.7570\) |
Rank
sage: E.rank()
The elliptic curves in class 303450t have rank \(1\).
Complex multiplication
The elliptic curves in class 303450t do not have complex multiplication.Modular form 303450.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
