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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 114240cc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
114240.et7 | 114240cc1 | \([0, -1, 0, -2068865, 1365924225]\) | \(-3735772816268612449/909650165760000\) | \(-238459333052989440000\) | \([2]\) | \(3538944\) | \(2.6285\) | \(\Gamma_0(N)\)-optimal |
114240.et6 | 114240cc2 | \([0, -1, 0, -34836865, 79150602625]\) | \(17836145204788591940449/770635366502400\) | \(202017437516405145600\) | \([2, 2]\) | \(7077888\) | \(2.9751\) | |
114240.et8 | 114240cc3 | \([0, -1, 0, 14888575, -9292637823]\) | \(1392333139184610040991/947901937500000000\) | \(-248486805504000000000000\) | \([2]\) | \(10616832\) | \(3.1778\) | |
114240.et5 | 114240cc4 | \([0, -1, 0, -36577665, 70804859265]\) | \(20645800966247918737249/3688936444974392640\) | \(967032555431367184220160\) | \([2]\) | \(14155776\) | \(3.3216\) | |
114240.et3 | 114240cc5 | \([0, -1, 0, -557384065, 5065191475585]\) | \(73054578035931991395831649/136386452160\) | \(35752890115031040\) | \([2]\) | \(14155776\) | \(3.3216\) | |
114240.et4 | 114240cc6 | \([0, -1, 0, -65111425, -77436637823]\) | \(116454264690812369959009/57505157319440250000\) | \(15074631960347344896000000\) | \([2, 2]\) | \(21233664\) | \(3.5244\) | |
114240.et1 | 114240cc7 | \([0, -1, 0, -851191425, -9551115581823]\) | \(260174968233082037895439009/223081361502731896500\) | \(58479440429772150276096000\) | \([2]\) | \(42467328\) | \(3.8709\) | |
114240.et2 | 114240cc8 | \([0, -1, 0, -559031425, 5033746306177]\) | \(73704237235978088924479009/899277423164136103500\) | \(235740180817939294715904000\) | \([2]\) | \(42467328\) | \(3.8709\) |
Rank
sage: E.rank()
The elliptic curves in class 114240cc have rank \(1\).
Complex multiplication
The elliptic curves in class 114240cc do not have complex multiplication.Modular form 114240.2.a.cc
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.