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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 62400cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 62400.fm6 | 62400cs1 | \([0, 1, 0, -176033, 28368063]\) | \(147281603041/5265\) | \(21565440000000\) | \([2]\) | \(294912\) | \(1.6476\) | \(\Gamma_0(N)\)-optimal |
| 62400.fm5 | 62400cs2 | \([0, 1, 0, -184033, 25640063]\) | \(168288035761/27720225\) | \(113542041600000000\) | \([2, 2]\) | \(589824\) | \(1.9941\) | |
| 62400.fm7 | 62400cs3 | \([0, 1, 0, 335967, 144720063]\) | \(1023887723039/2798036865\) | \(-11460758999040000000\) | \([2]\) | \(1179648\) | \(2.3407\) | |
| 62400.fm4 | 62400cs4 | \([0, 1, 0, -832033, -267903937]\) | \(15551989015681/1445900625\) | \(5922408960000000000\) | \([2, 2]\) | \(1179648\) | \(2.3407\) | |
| 62400.fm8 | 62400cs5 | \([0, 1, 0, 967967, -1266903937]\) | \(24487529386319/183539412225\) | \(-751777432473600000000\) | \([2]\) | \(2359296\) | \(2.6873\) | |
| 62400.fm2 | 62400cs6 | \([0, 1, 0, -13000033, -18045351937]\) | \(59319456301170001/594140625\) | \(2433600000000000000\) | \([2, 2]\) | \(2359296\) | \(2.6873\) | |
| 62400.fm3 | 62400cs7 | \([0, 1, 0, -12688033, -18952335937]\) | \(-55150149867714721/5950927734375\) | \(-24375000000000000000000\) | \([2]\) | \(4718592\) | \(3.0339\) | |
| 62400.fm1 | 62400cs8 | \([0, 1, 0, -208000033, -1154700351937]\) | \(242970740812818720001/24375\) | \(99840000000000\) | \([2]\) | \(4718592\) | \(3.0339\) |
Rank
sage: E.rank()
The elliptic curves in class 62400cs have rank \(1\).
Complex multiplication
The elliptic curves in class 62400cs do not have complex multiplication.Modular form 62400.2.a.cs
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
