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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 25872.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25872.bd1 | 25872bm4 | \([0, -1, 0, -177592, -28734992]\) | \(1285429208617/614922\) | \(296324949516288\) | \([2]\) | \(147456\) | \(1.7318\) | |
| 25872.bd2 | 25872bm3 | \([0, -1, 0, -99192, 11857392]\) | \(223980311017/4278582\) | \(2061807180668928\) | \([2]\) | \(147456\) | \(1.7318\) | |
| 25872.bd3 | 25872bm2 | \([0, -1, 0, -12952, -285200]\) | \(498677257/213444\) | \(102856594046976\) | \([2, 2]\) | \(73728\) | \(1.3852\) | |
| 25872.bd4 | 25872bm1 | \([0, -1, 0, 2728, -34320]\) | \(4657463/3696\) | \(-1781066563584\) | \([2]\) | \(36864\) | \(1.0386\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25872.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 25872.bd do not have complex multiplication.Modular form 25872.2.a.bd
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.
