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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 77b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
77.b2 | 77b1 | \([0, 1, 1, -49, 600]\) | \(-13278380032/156590819\) | \(-156590819\) | \([3]\) | \(20\) | \(0.25431\) | \(\Gamma_0(N)\)-optimal |
77.b3 | 77b2 | \([0, 1, 1, 441, -15815]\) | \(9463555063808/115539436859\) | \(-115539436859\) | \([]\) | \(60\) | \(0.80361\) | |
77.b1 | 77b3 | \([0, 1, 1, -89, 295]\) | \(-78843215872/539\) | \(-539\) | \([3]\) | \(60\) | \(-0.29500\) |
Rank
sage: E.rank()
The elliptic curves in class 77b have rank \(0\).
Complex multiplication
The elliptic curves in class 77b do not have complex multiplication.Modular form 77.2.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 Cremona numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.