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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 400554.ch
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
400554.ch1 | 400554ch4 | \([1, -1, 0, -104682501, -412222484093]\) | \(7209828390823479793/49509306\) | \(871179997203776106\) | \([2]\) | \(31457280\) | \(3.0421\) | |
400554.ch2 | 400554ch3 | \([1, -1, 0, -9121761, -899647025]\) | \(4770223741048753/2740574865798\) | \(48223944078768559530198\) | \([2]\) | \(31457280\) | \(3.0421\) | \(\Gamma_0(N)\)-optimal* |
400554.ch3 | 400554ch2 | \([1, -1, 0, -6546771, -6431240543]\) | \(1763535241378513/4612311396\) | \(81159558751848080196\) | \([2, 2]\) | \(15728640\) | \(2.6956\) | \(\Gamma_0(N)\)-optimal* |
400554.ch4 | 400554ch1 | \([1, -1, 0, -252351, -178363715]\) | \(-100999381393/723148272\) | \(-12724725116907829872\) | \([2]\) | \(7864320\) | \(2.3490\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 400554.ch have rank \(0\).
Complex multiplication
The elliptic curves in class 400554.ch do not have complex multiplication.Modular form 400554.2.a.ch
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.