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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 10725a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10725.h5 | 10725a1 | \([1, 1, 0, -600, 7875]\) | \(-1532808577/938223\) | \(-14659734375\) | \([2]\) | \(8192\) | \(0.65267\) | \(\Gamma_0(N)\)-optimal |
| 10725.h4 | 10725a2 | \([1, 1, 0, -10725, 423000]\) | \(8732907467857/1656369\) | \(25880765625\) | \([2, 2]\) | \(16384\) | \(0.99924\) | |
| 10725.h3 | 10725a3 | \([1, 1, 0, -11850, 327375]\) | \(11779205551777/3763454409\) | \(58803975140625\) | \([2, 2]\) | \(32768\) | \(1.3458\) | |
| 10725.h1 | 10725a4 | \([1, 1, 0, -171600, 27289125]\) | \(35765103905346817/1287\) | \(20109375\) | \([2]\) | \(32768\) | \(1.3458\) | |
| 10725.h2 | 10725a5 | \([1, 1, 0, -75225, -7721250]\) | \(3013001140430737/108679952667\) | \(1698124260421875\) | \([2]\) | \(65536\) | \(1.6924\) | |
| 10725.h6 | 10725a6 | \([1, 1, 0, 33525, 2278500]\) | \(266679605718863/296110251723\) | \(-4626722683171875\) | \([2]\) | \(65536\) | \(1.6924\) |
Rank
sage: E.rank()
The elliptic curves in class 10725a have rank \(1\).
Complex multiplication
The elliptic curves in class 10725a do not have complex multiplication.Modular form 10725.2.a.a
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
