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SageMath
E = EllipticCurve("gu1")
E.isogeny_class()
Elliptic curves in class 145200.gu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 145200.gu1 | 145200bp3 | \([0, 1, 0, -3897208, -2951226412]\) | \(57736239625/255552\) | \(28974461227008000000\) | \([2]\) | \(4976640\) | \(2.5867\) | |
| 145200.gu2 | 145200bp4 | \([0, 1, 0, -1961208, -5882330412]\) | \(-7357983625/127552392\) | \(-14461877959930368000000\) | \([2]\) | \(9953280\) | \(2.9333\) | |
| 145200.gu3 | 145200bp1 | \([0, 1, 0, -267208, 50057588]\) | \(18609625/1188\) | \(134695325952000000\) | \([2]\) | \(1658880\) | \(2.0374\) | \(\Gamma_0(N)\)-optimal |
| 145200.gu4 | 145200bp2 | \([0, 1, 0, 216792, 211713588]\) | \(9938375/176418\) | \(-20002255903872000000\) | \([2]\) | \(3317760\) | \(2.3840\) |
Rank
sage: E.rank()
The elliptic curves in class 145200.gu have rank \(2\).
Complex multiplication
The elliptic curves in class 145200.gu do not have complex multiplication.Modular form 145200.2.a.gu
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
