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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 27690bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27690.bg2 | 27690bc1 | \([1, 0, 0, -319040, 74649600]\) | \(-3591362198523471528961/330727587840000000\) | \(-330727587840000000\) | \([7]\) | \(526848\) | \(2.1043\) | \(\Gamma_0(N)\)-optimal |
27690.bg1 | 27690bc2 | \([1, 0, 0, -8462840, -15758963160]\) | \(-67030445471226692469644161/68484636225100842737640\) | \(-68484636225100842737640\) | \([]\) | \(3687936\) | \(3.0772\) |
Rank
sage: E.rank()
The elliptic curves in class 27690bc have rank \(1\).
Complex multiplication
The elliptic curves in class 27690bc do not have complex multiplication.Modular form 27690.2.a.bc
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.