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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 35904cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 35904.bw4 | 35904cr1 | \([0, 1, 0, -374209, 68778047]\) | \(22106889268753393/4969545596928\) | \(1302736560961093632\) | \([2]\) | \(516096\) | \(2.1888\) | \(\Gamma_0(N)\)-optimal |
| 35904.bw2 | 35904cr2 | \([0, 1, 0, -5617089, 5121865791]\) | \(74768347616680342513/5615307472896\) | \(1472019162174849024\) | \([2, 2]\) | \(1032192\) | \(2.5354\) | |
| 35904.bw3 | 35904cr3 | \([0, 1, 0, -5248449, 5823535167]\) | \(-60992553706117024753/20624795251201152\) | \(-5406666326330874789888\) | \([2]\) | \(2064384\) | \(2.8820\) | |
| 35904.bw1 | 35904cr4 | \([0, 1, 0, -89871809, 327901698111]\) | \(306234591284035366263793/1727485056\) | \(452849842520064\) | \([4]\) | \(2064384\) | \(2.8820\) |
Rank
sage: E.rank()
The elliptic curves in class 35904cr have rank \(0\).
Complex multiplication
The elliptic curves in class 35904cr do not have complex multiplication.Modular form 35904.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
