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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 27075i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27075.u2 | 27075i1 | \([0, -1, 1, 602, -25287]\) | \(20480/243\) | \(-285803727075\) | \([]\) | \(40500\) | \(0.87912\) | \(\Gamma_0(N)\)-optimal |
27075.u1 | 27075i2 | \([0, -1, 1, -75208, 8162193]\) | \(-102400/3\) | \(-1378297294921875\) | \([]\) | \(202500\) | \(1.6838\) |
Rank
sage: E.rank()
The elliptic curves in class 27075i have rank \(0\).
Complex multiplication
The elliptic curves in class 27075i do not have complex multiplication.Modular form 27075.2.a.i
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.