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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 163170.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
163170.d1 | 163170ee2 | \([1, -1, 0, -36045, 2642625]\) | \(20713044141847/3696300\) | \(924248726100\) | \([2]\) | \(466944\) | \(1.3003\) | |
163170.d2 | 163170ee1 | \([1, -1, 0, -2025, 50301]\) | \(-3673650007/2157840\) | \(-539561418480\) | \([2]\) | \(233472\) | \(0.95374\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 163170.d have rank \(2\).
Complex multiplication
The elliptic curves in class 163170.d do not have complex multiplication.Modular form 163170.2.a.d
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.