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SageMath
E = EllipticCurve("gz1")
E.isogeny_class()
Elliptic curves in class 277200.gz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
277200.gz1 | 277200gz3 | \([0, 0, 0, -2070063075, 36251291097250]\) | \(21026497979043461623321/161783881875\) | \(7548188792760000000000\) | \([2]\) | \(94371840\) | \(3.7905\) | |
277200.gz2 | 277200gz2 | \([0, 0, 0, -129465075, 565634475250]\) | \(5143681768032498601/14238434358225\) | \(664308393417345600000000\) | \([2, 2]\) | \(47185920\) | \(3.4439\) | |
277200.gz3 | 277200gz4 | \([0, 0, 0, -78435075, 1016076285250]\) | \(-1143792273008057401/8897444448004035\) | \(-415119168166076256960000000\) | \([2]\) | \(94371840\) | \(3.7905\) | |
277200.gz4 | 277200gz1 | \([0, 0, 0, -11367075, 1007937250]\) | \(3481467828171481/2005331497785\) | \(93560746360656960000000\) | \([2]\) | \(23592960\) | \(3.0973\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 277200.gz have rank \(0\).
Complex multiplication
The elliptic curves in class 277200.gz do not have complex multiplication.Modular form 277200.2.a.gz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.