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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 175329p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 175329.o2 | 175329p1 | \([0, 0, 1, -18084, 383292]\) | \(7414712369152/3571421301\) | \(315031501539909\) | \([]\) | \(497664\) | \(1.4757\) | \(\Gamma_0(N)\)-optimal |
| 175329.o1 | 175329p2 | \([0, 0, 1, -766524, -258296283]\) | \(564661380021747712/27978783021\) | \(2467980471499389\) | \([]\) | \(1492992\) | \(2.0250\) |
Rank
sage: E.rank()
The elliptic curves in class 175329p have rank \(2\).
Complex multiplication
The elliptic curves in class 175329p do not have complex multiplication.Modular form 175329.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
