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SageMath
E = EllipticCurve("ih1")
E.isogeny_class()
Elliptic curves in class 78400.ih
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
78400.ih1 | 78400ek2 | \([0, 1, 0, -393633, 94928063]\) | \(-349938025/8\) | \(-154204897280000\) | \([]\) | \(414720\) | \(1.8359\) | |
78400.ih2 | 78400ek3 | \([0, 1, 0, -236833, -53561537]\) | \(-121945/32\) | \(-385512243200000000\) | \([]\) | \(691200\) | \(2.0913\) | |
78400.ih3 | 78400ek1 | \([0, 1, 0, -1633, 299263]\) | \(-25/2\) | \(-38551224320000\) | \([]\) | \(138240\) | \(1.2866\) | \(\Gamma_0(N)\)-optimal |
78400.ih4 | 78400ek4 | \([0, 1, 0, 1723167, 395278463]\) | \(46969655/32768\) | \(-394764537036800000000\) | \([]\) | \(2073600\) | \(2.6406\) |
Rank
sage: E.rank()
The elliptic curves in class 78400.ih have rank \(1\).
Complex multiplication
The elliptic curves in class 78400.ih do not have complex multiplication.Modular form 78400.2.a.ih
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.
\(\left(\begin{array}{rrrr} 1 & 15 & 3 & 5 \\ 15 & 1 & 5 & 3 \\ 3 & 5 & 1 & 15 \\ 5 & 3 & 15 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.