Show commands:
SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 187395h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187395.g6 | 187395h1 | \([1, 1, 1, -105730, -13276258]\) | \(147281603041/5265\) | \(4672706880465\) | \([2]\) | \(691200\) | \(1.5201\) | \(\Gamma_0(N)\)-optimal |
| 187395.g5 | 187395h2 | \([1, 1, 1, -110535, -12009660]\) | \(168288035761/27720225\) | \(24601801725648225\) | \([2, 2]\) | \(1382400\) | \(1.8667\) | |
| 187395.g4 | 187395h3 | \([1, 1, 1, -499740, 124367772]\) | \(15551989015681/1445900625\) | \(1283242127047700625\) | \([2, 2]\) | \(2764800\) | \(2.2133\) | |
| 187395.g7 | 187395h4 | \([1, 1, 1, 201790, -67228720]\) | \(1023887723039/2798036865\) | \(-2483268017261200065\) | \([2]\) | \(2764800\) | \(2.2133\) | |
| 187395.g2 | 187395h5 | \([1, 1, 1, -7808145, 8394558870]\) | \(59319456301170001/594140625\) | \(527301991719140625\) | \([2, 2]\) | \(5529600\) | \(2.5598\) | |
| 187395.g8 | 187395h6 | \([1, 1, 1, 581385, 590116422]\) | \(24487529386319/183539412225\) | \(-162891903958263900225\) | \([2]\) | \(5529600\) | \(2.5598\) | |
| 187395.g1 | 187395h7 | \([1, 1, 1, -124930020, 537410643870]\) | \(242970740812818720001/24375\) | \(21632902224375\) | \([2]\) | \(11059200\) | \(2.9064\) | |
| 187395.g3 | 187395h8 | \([1, 1, 1, -7620750, 8816872242]\) | \(-55150149867714721/5950927734375\) | \(-5281470269622802734375\) | \([2]\) | \(11059200\) | \(2.9064\) |
Rank
sage: E.rank()
The elliptic curves in class 187395h have rank \(1\).
Complex multiplication
The elliptic curves in class 187395h do not have complex multiplication.Modular form 187395.2.a.h
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
