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SageMath
E = EllipticCurve("hz1")
E.isogeny_class()
Elliptic curves in class 248430.hz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 248430.hz1 | 248430hz3 | \([1, 0, 0, -1125334246, 14530066897436]\) | \(277536408914951281369/2063880\) | \(1172013991902379080\) | \([2]\) | \(74317824\) | \(3.5181\) | |
| 248430.hz2 | 248430hz4 | \([1, 0, 0, -75303446, 193099387116]\) | \(83161039719198169/19757817763320\) | \(11219857195218911964936120\) | \([2]\) | \(74317824\) | \(3.5181\) | |
| 248430.hz3 | 248430hz2 | \([1, 0, 0, -70334846, 227018031876]\) | \(67762119444423769/5843073600\) | \(3318101834852513217600\) | \([2, 2]\) | \(37158912\) | \(3.1715\) | |
| 248430.hz4 | 248430hz1 | \([1, 0, 0, -4086846, 4067012676]\) | \(-13293525831769/4892160000\) | \(-2778107240064898560000\) | \([2]\) | \(18579456\) | \(2.8249\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 248430.hz have rank \(0\).
Complex multiplication
The elliptic curves in class 248430.hz do not have complex multiplication.Modular form 248430.2.a.hz
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
