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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 250173.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 250173.j1 | 250173j6 | \([1, -1, 1, -14682299, -21650372260]\) | \(10206027697760497/5557167\) | \(190591084869383583\) | \([2]\) | \(8847360\) | \(2.6443\) | |
| 250173.j2 | 250173j4 | \([1, -1, 1, -922784, -334131622]\) | \(2533811507137/58110129\) | \(1992970973881085121\) | \([2, 2]\) | \(4423680\) | \(2.2977\) | |
| 250173.j3 | 250173j2 | \([1, -1, 1, -126779, 9742538]\) | \(6570725617/2614689\) | \(89674543361040561\) | \([2, 2]\) | \(2211840\) | \(1.9512\) | |
| 250173.j4 | 250173j1 | \([1, -1, 1, -110534, 14167676]\) | \(4354703137/1617\) | \(55457355201633\) | \([2]\) | \(1105920\) | \(1.6046\) | \(\Gamma_0(N)\)-optimal |
| 250173.j5 | 250173j5 | \([1, -1, 1, 100651, -1035389284]\) | \(3288008303/13504609503\) | \(-463160127437983607247\) | \([2]\) | \(8847360\) | \(2.6443\) | |
| 250173.j6 | 250173j3 | \([1, -1, 1, 409306, 70212926]\) | \(221115865823/190238433\) | \(-6524502382116920817\) | \([2]\) | \(4423680\) | \(2.2977\) |
Rank
sage: E.rank()
The elliptic curves in class 250173.j have rank \(1\).
Complex multiplication
The elliptic curves in class 250173.j do not have complex multiplication.Modular form 250173.2.a.j
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
