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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 1617g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1617.e4 | 1617g1 | \([1, 0, 0, -1667, -26328]\) | \(4354703137/1617\) | \(190238433\) | \([2]\) | \(960\) | \(0.55602\) | \(\Gamma_0(N)\)-optimal |
1617.e3 | 1617g2 | \([1, 0, 0, -1912, -18145]\) | \(6570725617/2614689\) | \(307615546161\) | \([2, 2]\) | \(1920\) | \(0.90260\) | |
1617.e2 | 1617g3 | \([1, 0, 0, -13917, 618120]\) | \(2533811507137/58110129\) | \(6836598566721\) | \([2, 2]\) | \(3840\) | \(1.2492\) | |
1617.e6 | 1617g4 | \([1, 0, 0, 6173, -129718]\) | \(221115865823/190238433\) | \(-22381361404017\) | \([2]\) | \(3840\) | \(1.2492\) | |
1617.e1 | 1617g5 | \([1, 0, 0, -221432, 40087473]\) | \(10206027697760497/5557167\) | \(653795140383\) | \([4]\) | \(7680\) | \(1.5957\) | |
1617.e5 | 1617g6 | \([1, 0, 0, 1518, 1917747]\) | \(3288008303/13504609503\) | \(-1588803803418447\) | \([2]\) | \(7680\) | \(1.5957\) |
Rank
sage: E.rank()
The elliptic curves in class 1617g have rank \(1\).
Complex multiplication
The elliptic curves in class 1617g do not have complex multiplication.Modular form 1617.2.a.g
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.