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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 34320.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34320.q1 | 34320bm4 | \([0, -1, 0, -36345160, 84349085680]\) | \(1296294060988412126189641/647824320\) | \(2653488414720\) | \([2]\) | \(995328\) | \(2.6200\) | |
| 34320.q2 | 34320bm3 | \([0, -1, 0, -2271560, 1318537200]\) | \(-316472948332146183241/7074906009600\) | \(-28978815015321600\) | \([2]\) | \(497664\) | \(2.2734\) | |
| 34320.q3 | 34320bm2 | \([0, -1, 0, -449560, 115375600]\) | \(2453170411237305241/19353090685500\) | \(79270259447808000\) | \([2]\) | \(331776\) | \(2.0707\) | |
| 34320.q4 | 34320bm1 | \([0, -1, 0, -9560, 4143600]\) | \(-23592983745241/1794399750000\) | \(-7349861376000000\) | \([2]\) | \(165888\) | \(1.7241\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 34320.q have rank \(1\).
Complex multiplication
The elliptic curves in class 34320.q do not have complex multiplication.Modular form 34320.2.a.q
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.
