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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 47040.ew
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.ew1 | 47040cf8 | \([0, 1, 0, -20230401, -22041034401]\) | \(29689921233686449/10380965400750\) | \(320159140657977556992000\) | \([2]\) | \(5308416\) | \(3.2117\) | |
| 47040.ew2 | 47040cf5 | \([0, 1, 0, -18066561, -29563087905]\) | \(21145699168383889/2593080\) | \(79973127007764480\) | \([2]\) | \(1769472\) | \(2.6624\) | |
| 47040.ew3 | 47040cf6 | \([0, 1, 0, -8470401, 9233509599]\) | \(2179252305146449/66177562500\) | \(2040980845510656000000\) | \([2, 2]\) | \(2654208\) | \(2.8651\) | |
| 47040.ew4 | 47040cf3 | \([0, 1, 0, -8407681, 9380638175]\) | \(2131200347946769/2058000\) | \(63470735720448000\) | \([2]\) | \(1327104\) | \(2.5186\) | |
| 47040.ew5 | 47040cf2 | \([0, 1, 0, -1132161, -459628065]\) | \(5203798902289/57153600\) | \(1762673003436441600\) | \([2, 2]\) | \(884736\) | \(2.3158\) | |
| 47040.ew6 | 47040cf4 | \([0, 1, 0, -254081, -1153486881]\) | \(-58818484369/18600435000\) | \(-573655633707663360000\) | \([2]\) | \(1769472\) | \(2.6624\) | |
| 47040.ew7 | 47040cf1 | \([0, 1, 0, -128641, 6205919]\) | \(7633736209/3870720\) | \(119376795999928320\) | \([2]\) | \(442368\) | \(1.9692\) | \(\Gamma_0(N)\)-optimal |
| 47040.ew8 | 47040cf7 | \([0, 1, 0, 2286079, 31092828255]\) | \(42841933504271/13565917968750\) | \(-418386197376000000000000\) | \([2]\) | \(5308416\) | \(3.2117\) |
Rank
sage: E.rank()
The elliptic curves in class 47040.ew have rank \(1\).
Complex multiplication
The elliptic curves in class 47040.ew do not have complex multiplication.Modular form 47040.2.a.ew
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.
