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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 16830z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16830.t4 | 16830z1 | \([1, -1, 0, -1494, -123660]\) | \(-506071034209/8823767040\) | \(-6432526172160\) | \([2]\) | \(40960\) | \(1.1388\) | \(\Gamma_0(N)\)-optimal |
| 16830.t3 | 16830z2 | \([1, -1, 0, -47574, -3966732]\) | \(16334668434139489/72511718400\) | \(52861042713600\) | \([2, 2]\) | \(81920\) | \(1.4854\) | |
| 16830.t1 | 16830z3 | \([1, -1, 0, -760374, -255014892]\) | \(66692696957462376289/1322972640\) | \(964447054560\) | \([2]\) | \(163840\) | \(1.8320\) | |
| 16830.t2 | 16830z4 | \([1, -1, 0, -72054, 571860]\) | \(56751044592329569/32660264340000\) | \(23809332703860000\) | \([2]\) | \(163840\) | \(1.8320\) |
Rank
sage: E.rank()
The elliptic curves in class 16830z have rank \(1\).
Complex multiplication
The elliptic curves in class 16830z do not have complex multiplication.Modular form 16830.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
