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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 40560.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40560.q1 | 40560bv7 | \([0, -1, 0, -14421840, 21085221312]\) | \(16778985534208729/81000\) | \(1601419382784000\) | \([2]\) | \(1327104\) | \(2.5398\) | |
| 40560.q2 | 40560bv8 | \([0, -1, 0, -1226320, 71550400]\) | \(10316097499609/5859375000\) | \(115843416000000000000\) | \([2]\) | \(1327104\) | \(2.5398\) | |
| 40560.q3 | 40560bv6 | \([0, -1, 0, -901840, 329317312]\) | \(4102915888729/9000000\) | \(177935486976000000\) | \([2, 2]\) | \(663552\) | \(2.1932\) | |
| 40560.q4 | 40560bv5 | \([0, -1, 0, -780160, -264967808]\) | \(2656166199049/33750\) | \(667258076160000\) | \([2]\) | \(442368\) | \(1.9905\) | |
| 40560.q5 | 40560bv4 | \([0, -1, 0, -185280, 26501760]\) | \(35578826569/5314410\) | \(105069125704458240\) | \([2]\) | \(442368\) | \(1.9905\) | |
| 40560.q6 | 40560bv2 | \([0, -1, 0, -50080, -3891200]\) | \(702595369/72900\) | \(1441277444505600\) | \([2, 2]\) | \(221184\) | \(1.6439\) | |
| 40560.q7 | 40560bv3 | \([0, -1, 0, -36560, 8817600]\) | \(-273359449/1536000\) | \(-30367656443904000\) | \([2]\) | \(331776\) | \(1.8466\) | |
| 40560.q8 | 40560bv1 | \([0, -1, 0, 4000, -300288]\) | \(357911/2160\) | \(-42704516874240\) | \([2]\) | \(110592\) | \(1.2973\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 40560.q have rank \(1\).
Complex multiplication
The elliptic curves in class 40560.q do not have complex multiplication.Modular form 40560.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
