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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 49005.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
49005.i1 | 49005h2 | \([0, 0, 1, -19602, 1051157]\) | \(884736/5\) | \(4707400747005\) | \([]\) | \(97200\) | \(1.2736\) | |
49005.i2 | 49005h1 | \([0, 0, 1, -1452, -20298]\) | \(2359296/125\) | \(17937055125\) | \([]\) | \(32400\) | \(0.72431\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 49005.i have rank \(1\).
Complex multiplication
The elliptic curves in class 49005.i do not have complex multiplication.Modular form 49005.2.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 LMFDB 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 LMFDB labels.