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SageMath
E = EllipticCurve("cx1")
E.isogeny_class()
Elliptic curves in class 303450.cx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 303450.cx1 | 303450cx7 | \([1, 0, 1, -46608626, 77049510398]\) | \(29689921233686449/10380965400750\) | \(3915176072612746511718750\) | \([2]\) | \(63700992\) | \(3.4204\) | |
| 303450.cx2 | 303450cx4 | \([1, 0, 1, -41623376, 103356891398]\) | \(21145699168383889/2593080\) | \(977978865976875000\) | \([2]\) | \(21233664\) | \(2.8710\) | |
| 303450.cx3 | 303450cx6 | \([1, 0, 1, -19514876, -32300864602]\) | \(2179252305146449/66177562500\) | \(24958835642118164062500\) | \([2, 2]\) | \(31850496\) | \(3.0738\) | |
| 303450.cx4 | 303450cx3 | \([1, 0, 1, -19370376, -32815284602]\) | \(2131200347946769/2058000\) | \(776173703156250000\) | \([2]\) | \(15925248\) | \(2.7272\) | |
| 303450.cx5 | 303450cx2 | \([1, 0, 1, -2608376, 1605771398]\) | \(5203798902289/57153600\) | \(21555452556225000000\) | \([2, 2]\) | \(10616832\) | \(2.5245\) | |
| 303450.cx6 | 303450cx5 | \([1, 0, 1, -585376, 4033371398]\) | \(-58818484369/18600435000\) | \(-7015145050664296875000\) | \([2]\) | \(21233664\) | \(2.8710\) | |
| 303450.cx7 | 303450cx1 | \([1, 0, 1, -296376, -21876602]\) | \(7633736209/3870720\) | \(1459840173120000000\) | \([2]\) | \(5308416\) | \(2.1779\) | \(\Gamma_0(N)\)-optimal |
| 303450.cx8 | 303450cx8 | \([1, 0, 1, 5266874, -108727781602]\) | \(42841933504271/13565917968750\) | \(-5116379390922546386718750\) | \([2]\) | \(63700992\) | \(3.4204\) |
Rank
sage: E.rank()
The elliptic curves in class 303450.cx have rank \(1\).
Complex multiplication
The elliptic curves in class 303450.cx do not have complex multiplication.Modular form 303450.2.a.cx
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
