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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1856j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1856.i1 | 1856j1 | \([0, 1, 0, -17, -49]\) | \(-35152/29\) | \(-475136\) | \([]\) | \(256\) | \(-0.21275\) | \(\Gamma_0(N)\)-optimal |
1856.i2 | 1856j2 | \([0, 1, 0, 143, 751]\) | \(19600688/24389\) | \(-399589376\) | \([]\) | \(768\) | \(0.33656\) |
Rank
sage: E.rank()
The elliptic curves in class 1856j have rank \(1\).
Complex multiplication
The elliptic curves in class 1856j do not have complex multiplication.Modular form 1856.2.a.j
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.