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SageMath
E = EllipticCurve("cu1")
E.isogeny_class()
Elliptic curves in class 48510.cu
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
48510.cu1 | 48510dk8 | \([1, -1, 1, -11231382938, -458135277511719]\) | \(1826870018430810435423307849/7641104625000000000\) | \(655347903841409625000000000\) | \([2]\) | \(63700992\) | \(4.3545\) | |
48510.cu2 | 48510dk6 | \([1, -1, 1, -712862618, -6924415048743]\) | \(467116778179943012100169/28800309694464000000\) | \(2470090846092872454144000000\) | \([2, 2]\) | \(31850496\) | \(4.0079\) | |
48510.cu3 | 48510dk5 | \([1, -1, 1, -193058123, -90611800653]\) | \(9278380528613437145689/5328033205714065000\) | \(456964740613290390191865000\) | \([2]\) | \(21233664\) | \(3.8052\) | |
48510.cu4 | 48510dk3 | \([1, -1, 1, -134835098, 469250565081]\) | \(3160944030998056790089/720291785342976000\) | \(61776632417031706116096000\) | \([2]\) | \(15925248\) | \(3.6614\) | |
48510.cu5 | 48510dk2 | \([1, -1, 1, -126502403, 545474526531]\) | \(2610383204210122997209/12104550027662400\) | \(1038160302323046745550400\) | \([2, 2]\) | \(10616832\) | \(3.4586\) | |
48510.cu6 | 48510dk1 | \([1, -1, 1, -126361283, 546756855747]\) | \(2601656892010848045529/56330588160\) | \(4831256040131727360\) | \([2]\) | \(5308416\) | \(3.1120\) | \(\Gamma_0(N)\)-optimal |
48510.cu7 | 48510dk4 | \([1, -1, 1, -62204603, 1099490090451]\) | \(-310366976336070130009/5909282337130963560\) | \(-506816223949537013533550760\) | \([2]\) | \(21233664\) | \(3.8052\) | |
48510.cu8 | 48510dk7 | \([1, -1, 1, 557217382, -28915088200743]\) | \(223090928422700449019831/4340371122724101696000\) | \(-372256794896461155675441216000\) | \([2]\) | \(63700992\) | \(4.3545\) |
Rank
sage: E.rank()
The elliptic curves in class 48510.cu have rank \(1\).
Complex multiplication
The elliptic curves in class 48510.cu do not have complex multiplication.Modular form 48510.2.a.cu
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.