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SageMath
E = EllipticCurve("hd1")
E.isogeny_class()
Elliptic curves in class 414050.hd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414050.hd1 | 414050hd2 | \([1, 1, 1, -2074563, -1151016469]\) | \(-5452947409/250\) | \(-45270189097656250\) | \([]\) | \(8864640\) | \(2.2716\) | |
414050.hd2 | 414050hd1 | \([1, 1, 1, -4313, -4097969]\) | \(-49/40\) | \(-7243230255625000\) | \([]\) | \(2954880\) | \(1.7223\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 414050.hd have rank \(0\).
Complex multiplication
The elliptic curves in class 414050.hd do not have complex multiplication.Modular form 414050.2.a.hd
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.