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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 25410.cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25410.cr1 | 25410cr4 | \([1, 0, 0, -3151266, -2153256984]\) | \(1953542217204454969/170843779260\) | \(302660176429624860\) | \([2]\) | \(614400\) | \(2.3956\) | |
| 25410.cr2 | 25410cr3 | \([1, 0, 0, -1142666, 446011776]\) | \(93137706732176569/5369647977540\) | \(9512658940738739940\) | \([2]\) | \(614400\) | \(2.3956\) | |
| 25410.cr3 | 25410cr2 | \([1, 0, 0, -210966, -28596204]\) | \(586145095611769/140040608400\) | \(248090480257712400\) | \([2, 2]\) | \(307200\) | \(2.0490\) | |
| 25410.cr4 | 25410cr1 | \([1, 0, 0, 31034, -2799004]\) | \(1865864036231/2993760000\) | \(-5303628459360000\) | \([2]\) | \(153600\) | \(1.7024\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25410.cr have rank \(1\).
Complex multiplication
The elliptic curves in class 25410.cr do not have complex multiplication.Modular form 25410.2.a.cr
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
