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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 49725h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
49725.u4 | 49725h1 | \([1, -1, 0, 51061833, -221933622384]\) | \(1292603583867446566871/2615843353271484375\) | \(-29796090695858001708984375\) | \([2]\) | \(11501568\) | \(3.5725\) | \(\Gamma_0(N)\)-optimal |
49725.u3 | 49725h2 | \([1, -1, 0, -388391292, -2377451200509]\) | \(568832774079017834683129/114800389711906640625\) | \(1307648189062186578369140625\) | \([2, 2]\) | \(23003136\) | \(3.9190\) | |
49725.u2 | 49725h3 | \([1, -1, 0, -1933156917, 30595571065116]\) | \(70141892778055497175333129/5090453819946781723125\) | \(57983450542831310564970703125\) | \([2]\) | \(46006272\) | \(4.2656\) | |
49725.u1 | 49725h4 | \([1, -1, 0, -5874875667, -173308871903634]\) | \(1968666709544018637994033129/113621848881699526875\) | \(1294223872418108673310546875\) | \([2]\) | \(46006272\) | \(4.2656\) |
Rank
sage: E.rank()
The elliptic curves in class 49725h have rank \(1\).
Complex multiplication
The elliptic curves in class 49725h do not have complex multiplication.Modular form 49725.2.a.h
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.