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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 309738bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 309738.bp4 | 309738bp1 | \([1, 1, 1, 129772, -12884011]\) | \(5137417856375/4510142208\) | \(-212183613610645248\) | \([2]\) | \(3981312\) | \(2.0122\) | \(\Gamma_0(N)\)-optimal |
| 309738.bp3 | 309738bp2 | \([1, 1, 1, -649988, -115188523]\) | \(645532578015625/252306960048\) | \(11870003217889962288\) | \([2]\) | \(7962624\) | \(2.3587\) | |
| 309738.bp2 | 309738bp3 | \([1, 1, 1, -1348523, 804317465]\) | \(-5764706497797625/2612665516032\) | \(-122915150960045064192\) | \([2]\) | \(11943936\) | \(2.5615\) | |
| 309738.bp1 | 309738bp4 | \([1, 1, 1, -23528363, 43913054489]\) | \(30618029936661765625/3678951124992\) | \(173079496831189757952\) | \([2]\) | \(23887872\) | \(2.9080\) |
Rank
sage: E.rank()
The elliptic curves in class 309738bp have rank \(1\).
Complex multiplication
The elliptic curves in class 309738bp do not have complex multiplication.Modular form 309738.2.a.bp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
