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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 323400.cz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 323400.cz1 | 323400cz4 | \([0, -1, 0, -36476008, -84766747988]\) | \(1425631925916578/270703125\) | \(1019134462500000000000\) | \([2]\) | \(18874368\) | \(3.0322\) | |
| 323400.cz2 | 323400cz3 | \([0, -1, 0, -15994008, 23846064012]\) | \(120186986927618/4332064275\) | \(16309216956463200000000\) | \([2]\) | \(18874368\) | \(3.0322\) | |
| 323400.cz3 | 323400cz2 | \([0, -1, 0, -2519008, -1028785988]\) | \(939083699236/300155625\) | \(565008146010000000000\) | \([2, 2]\) | \(9437184\) | \(2.6856\) | |
| 323400.cz4 | 323400cz1 | \([0, -1, 0, 445492, -109790988]\) | \(20777545136/23059575\) | \(-10851743756700000000\) | \([2]\) | \(4718592\) | \(2.3391\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 323400.cz have rank \(1\).
Complex multiplication
The elliptic curves in class 323400.cz do not have complex multiplication.Modular form 323400.2.a.cz
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.
