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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 221991e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221991.h4 | 221991e1 | \([1, 0, 0, -32694, -2277357]\) | \(4354703137/1617\) | \(1435093452177\) | \([2]\) | \(576000\) | \(1.3001\) | \(\Gamma_0(N)\)-optimal |
221991.h3 | 221991e2 | \([1, 0, 0, -37499, -1565256]\) | \(6570725617/2614689\) | \(2320546112170209\) | \([2, 2]\) | \(1152000\) | \(1.6466\) | |
221991.h2 | 221991e3 | \([1, 0, 0, -272944, 53764319]\) | \(2533811507137/58110129\) | \(51572953390884849\) | \([2, 2]\) | \(2304000\) | \(1.9932\) | |
221991.h6 | 221991e4 | \([1, 0, 0, 121066, -11301147]\) | \(221115865823/190238433\) | \(-168837309555171873\) | \([2]\) | \(2304000\) | \(1.9932\) | |
221991.h1 | 221991e5 | \([1, 0, 0, -4342779, 3483007290]\) | \(10206027697760497/5557167\) | \(4932006168431727\) | \([2]\) | \(4608000\) | \(2.3398\) | |
221991.h5 | 221991e6 | \([1, 0, 0, 29771, 166555928]\) | \(3288008303/13504609503\) | \(-11985390644380080543\) | \([2]\) | \(4608000\) | \(2.3398\) |
Rank
sage: E.rank()
The elliptic curves in class 221991e have rank \(1\).
Complex multiplication
The elliptic curves in class 221991e do not have complex multiplication.Modular form 221991.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.