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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 132600.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 132600.p1 | 132600ck6 | \([0, -1, 0, -4862008, -4092919988]\) | \(397210600760070242/3536192675535\) | \(113158165617120000000\) | \([2]\) | \(4325376\) | \(2.6711\) | |
| 132600.p2 | 132600ck4 | \([0, -1, 0, -527008, 42670012]\) | \(1011710313226084/536724738225\) | \(8587595811600000000\) | \([2, 2]\) | \(2162688\) | \(2.3245\) | |
| 132600.p3 | 132600ck2 | \([0, -1, 0, -414508, 102745012]\) | \(1969080716416336/2472575625\) | \(9890302500000000\) | \([2, 2]\) | \(1081344\) | \(1.9779\) | |
| 132600.p4 | 132600ck1 | \([0, -1, 0, -414383, 102810012]\) | \(31476797652269056/49725\) | \(12431250000\) | \([2]\) | \(540672\) | \(1.6314\) | \(\Gamma_0(N)\)-optimal |
| 132600.p5 | 132600ck3 | \([0, -1, 0, -304008, 158658012]\) | \(-194204905090564/566398828125\) | \(-9062381250000000000\) | \([2]\) | \(2162688\) | \(2.3245\) | |
| 132600.p6 | 132600ck5 | \([0, -1, 0, 2007992, 331660012]\) | \(27980756504588158/17683545112935\) | \(-565873443613920000000\) | \([2]\) | \(4325376\) | \(2.6711\) |
Rank
sage: E.rank()
The elliptic curves in class 132600.p have rank \(1\).
Complex multiplication
The elliptic curves in class 132600.p do not have complex multiplication.Modular form 132600.2.a.p
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
