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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 735a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
735.f3 | 735a1 | \([1, 1, 0, -123, -552]\) | \(1771561/105\) | \(12353145\) | \([2]\) | \(192\) | \(0.11385\) | \(\Gamma_0(N)\)-optimal |
735.f2 | 735a2 | \([1, 1, 0, -368, 1947]\) | \(47045881/11025\) | \(1297080225\) | \([2, 2]\) | \(384\) | \(0.46042\) | |
735.f1 | 735a3 | \([1, 1, 0, -5513, 155268]\) | \(157551496201/13125\) | \(1544143125\) | \([2]\) | \(768\) | \(0.80699\) | |
735.f4 | 735a4 | \([1, 1, 0, 857, 13462]\) | \(590589719/972405\) | \(-114402475845\) | \([2]\) | \(768\) | \(0.80699\) |
Rank
sage: E.rank()
The elliptic curves in class 735a have rank \(0\).
Complex multiplication
The elliptic curves in class 735a do not have complex multiplication.Modular form 735.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.