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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 450840bl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450840.bl4 | 450840bl1 | \([0, 1, 0, -5876, 613440]\) | \(-3631696/24375\) | \(-150618430560000\) | \([2]\) | \(1769472\) | \(1.4038\) | \(\Gamma_0(N)\)-optimal* |
450840.bl3 | 450840bl2 | \([0, 1, 0, -150376, 22346240]\) | \(15214885924/38025\) | \(939859006694400\) | \([2, 2]\) | \(3538944\) | \(1.7504\) | \(\Gamma_0(N)\)-optimal* |
450840.bl1 | 450840bl3 | \([0, 1, 0, -2404576, 1434377120]\) | \(31103978031362/195\) | \(9639579555840\) | \([2]\) | \(7077888\) | \(2.0970\) | \(\Gamma_0(N)\)-optimal* |
450840.bl2 | 450840bl4 | \([0, 1, 0, -208176, 3526560]\) | \(20183398562/11567205\) | \(571810219672872960\) | \([2]\) | \(7077888\) | \(2.0970\) |
Rank
sage: E.rank()
The elliptic curves in class 450840bl have rank \(0\).
Complex multiplication
The elliptic curves in class 450840bl do not have complex multiplication.Modular form 450840.2.a.bl
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.