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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 3174.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3174.b1 | 3174a2 | \([1, 1, 0, -16674, -835050]\) | \(3463512697/3174\) | \(469865911686\) | \([2]\) | \(8448\) | \(1.1632\) | |
3174.b2 | 3174a1 | \([1, 1, 0, -804, -19332]\) | \(-389017/828\) | \(-122573716092\) | \([2]\) | \(4224\) | \(0.81662\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3174.b have rank \(0\).
Complex multiplication
The elliptic curves in class 3174.b do not have complex multiplication.Modular form 3174.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.