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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 8085.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8085.b1 | 8085j2 | \([0, -1, 1, -1312726, 10945050906]\) | \(-2126464142970105856/438611057788643355\) | \(-51602152337776102072395\) | \([]\) | \(1440000\) | \(3.0374\) | |
8085.b2 | 8085j1 | \([0, -1, 1, -438076, -130549434]\) | \(-79028701534867456/16987307596875\) | \(-1998539751464746875\) | \([]\) | \(288000\) | \(2.2327\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8085.b have rank \(1\).
Complex multiplication
The elliptic curves in class 8085.b do not have complex multiplication.Modular form 8085.2.a.b
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.