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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 25230.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25230.o1 | 25230p7 | \([1, 1, 1, -4485491, -3658348087]\) | \(16778985534208729/81000\) | \(48180689001000\) | \([2]\) | \(580608\) | \(2.2478\) | |
| 25230.o2 | 25230p8 | \([1, 1, 1, -381411, -12498711]\) | \(10316097499609/5859375000\) | \(3485292896484375000\) | \([2]\) | \(580608\) | \(2.2478\) | |
| 25230.o3 | 25230p6 | \([1, 1, 1, -280491, -57186087]\) | \(4102915888729/9000000\) | \(5353409889000000\) | \([2, 2]\) | \(290304\) | \(1.9012\) | |
| 25230.o4 | 25230p5 | \([1, 1, 1, -242646, 45903693]\) | \(2656166199049/33750\) | \(20075287083750\) | \([2]\) | \(193536\) | \(1.6985\) | |
| 25230.o5 | 25230p4 | \([1, 1, 1, -57626, -4610131]\) | \(35578826569/5314410\) | \(3161135005355610\) | \([2]\) | \(193536\) | \(1.6985\) | |
| 25230.o6 | 25230p2 | \([1, 1, 1, -15576, 671349]\) | \(702595369/72900\) | \(43362620100900\) | \([2, 2]\) | \(96768\) | \(1.3519\) | |
| 25230.o7 | 25230p3 | \([1, 1, 1, -11371, -1532071]\) | \(-273359449/1536000\) | \(-913648621056000\) | \([2]\) | \(145152\) | \(1.5547\) | |
| 25230.o8 | 25230p1 | \([1, 1, 1, 1244, 52373]\) | \(357911/2160\) | \(-1284818373360\) | \([2]\) | \(48384\) | \(1.0054\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25230.o have rank \(0\).
Complex multiplication
The elliptic curves in class 25230.o do not have complex multiplication.Modular form 25230.2.a.o
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
