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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 50715bb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
50715.bj3 | 50715bb1 | \([1, -1, 0, -35730, -2288385]\) | \(58818484369/7455105\) | \(639395437497705\) | \([2]\) | \(184320\) | \(1.5701\) | \(\Gamma_0(N)\)-optimal |
50715.bj2 | 50715bb2 | \([1, -1, 0, -143775, 18650736]\) | \(3832302404449/472410225\) | \(40516792518987225\) | \([2, 2]\) | \(368640\) | \(1.9167\) | |
50715.bj4 | 50715bb3 | \([1, -1, 0, 211230, 95970825]\) | \(12152722588271/53476250625\) | \(-4586450581730075625\) | \([2]\) | \(737280\) | \(2.2633\) | |
50715.bj1 | 50715bb4 | \([1, -1, 0, -2227500, 1280137851]\) | \(14251520160844849/264449745\) | \(22680828828089145\) | \([2]\) | \(737280\) | \(2.2633\) |
Rank
sage: E.rank()
The elliptic curves in class 50715bb have rank \(0\).
Complex multiplication
The elliptic curves in class 50715bb do not have complex multiplication.Modular form 50715.2.a.bb
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.