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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 25578i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25578.k2 | 25578i1 | \([1, -1, 0, -450, 1269652]\) | \(-117649/8118144\) | \(-696261720599424\) | \([]\) | \(84672\) | \(1.5270\) | \(\Gamma_0(N)\)-optimal |
25578.k1 | 25578i2 | \([1, -1, 0, -2871360, -1877509418]\) | \(-30526075007211889/103499257854\) | \(-8876729872516364334\) | \([]\) | \(592704\) | \(2.4999\) |
Rank
sage: E.rank()
The elliptic curves in class 25578i have rank \(1\).
Complex multiplication
The elliptic curves in class 25578i do not have complex multiplication.Modular form 25578.2.a.i
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.