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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 27225.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27225.z1 | 27225bh2 | \([0, 0, 1, -1348050, -602431844]\) | \(-196566176333824/421875\) | \(-581455810546875\) | \([]\) | \(248832\) | \(2.0806\) | |
27225.z2 | 27225bh1 | \([0, 0, 1, -11550, -1340969]\) | \(-123633664/492075\) | \(-678210057421875\) | \([]\) | \(82944\) | \(1.5312\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 27225.z have rank \(1\).
Complex multiplication
The elliptic curves in class 27225.z do not have complex multiplication.Modular form 27225.2.a.z
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.