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SageMath
E = EllipticCurve("hj1")
E.isogeny_class()
Elliptic curves in class 499800.hj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
499800.hj1 | 499800hj4 | \([0, 1, 0, -222175094808, -40308067092236112]\) | \(322159999717985454060440834/4250799\) | \(16003272049632000000\) | \([2]\) | \(849346560\) | \(4.6687\) | |
499800.hj2 | 499800hj3 | \([0, 1, 0, -13921664808, -626413772228112]\) | \(79260902459030376659234/842751810121431609\) | \(3172765046687241835751712000000\) | \([2]\) | \(849346560\) | \(4.6687\) | \(\Gamma_0(N)\)-optimal* |
499800.hj3 | 499800hj2 | \([0, 1, 0, -13885943808, -629816983340112]\) | \(157304700372188331121828/18069292138401\) | \(34013346412651827984000000\) | \([2, 2]\) | \(424673280\) | \(4.3221\) | \(\Gamma_0(N)\)-optimal* |
499800.hj4 | 499800hj1 | \([0, 1, 0, -865639308, -9894245486112]\) | \(-152435594466395827792/1646846627220711\) | \(-774999435383557713756000000\) | \([2]\) | \(212336640\) | \(3.9755\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 499800.hj have rank \(0\).
Complex multiplication
The elliptic curves in class 499800.hj do not have complex multiplication.Modular form 499800.2.a.hj
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.