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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 12675p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12675.h2 | 12675p1 | \([1, 1, 1, -426813, 133953906]\) | \(-51895117/16875\) | \(-2796108233115234375\) | \([2]\) | \(179712\) | \(2.2521\) | \(\Gamma_0(N)\)-optimal |
12675.h1 | 12675p2 | \([1, 1, 1, -7292438, 7576291406]\) | \(258840217117/18225\) | \(3019796891764453125\) | \([2]\) | \(359424\) | \(2.5987\) |
Rank
sage: E.rank()
The elliptic curves in class 12675p have rank \(0\).
Complex multiplication
The elliptic curves in class 12675p do not have complex multiplication.Modular form 12675.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.