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SageMath
E = EllipticCurve("gp1")
E.isogeny_class()
Elliptic curves in class 303600.gp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 303600.gp1 | 303600gp5 | \([0, 1, 0, -37241608, 87463956788]\) | \(89254274298475942657/17457\) | \(1117248000000\) | \([2]\) | \(8388608\) | \(2.6130\) | |
| 303600.gp2 | 303600gp3 | \([0, 1, 0, -2327608, 1366032788]\) | \(21790813729717297/304746849\) | \(19503798336000000\) | \([2, 2]\) | \(4194304\) | \(2.2664\) | |
| 303600.gp3 | 303600gp6 | \([0, 1, 0, -2261608, 1447212788]\) | \(-19989223566735457/2584262514273\) | \(-165392800913472000000\) | \([2]\) | \(8388608\) | \(2.6130\) | |
| 303600.gp4 | 303600gp4 | \([0, 1, 0, -563608, -141431212]\) | \(309368403125137/44372288367\) | \(2839826455488000000\) | \([2]\) | \(4194304\) | \(2.2664\) | |
| 303600.gp5 | 303600gp2 | \([0, 1, 0, -149608, 20028788]\) | \(5786435182177/627352209\) | \(40150541376000000\) | \([2, 2]\) | \(2097152\) | \(1.9198\) | |
| 303600.gp6 | 303600gp1 | \([0, 1, 0, 12392, 1560788]\) | \(3288008303/18259263\) | \(-1168592832000000\) | \([2]\) | \(1048576\) | \(1.5733\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 303600.gp have rank \(1\).
Complex multiplication
The elliptic curves in class 303600.gp do not have complex multiplication.Modular form 303600.2.a.gp
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
