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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 1344.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1344.g1 | 1344a5 | \([0, -1, 0, -50177, -4309503]\) | \(53297461115137/147\) | \(38535168\) | \([2]\) | \(2048\) | \(1.1139\) | |
1344.g2 | 1344a3 | \([0, -1, 0, -3137, -66495]\) | \(13027640977/21609\) | \(5664669696\) | \([2, 2]\) | \(1024\) | \(0.76735\) | |
1344.g3 | 1344a4 | \([0, -1, 0, -2497, 48577]\) | \(6570725617/45927\) | \(12039487488\) | \([2]\) | \(1024\) | \(0.76735\) | |
1344.g4 | 1344a6 | \([0, -1, 0, -2177, -108927]\) | \(-4354703137/17294403\) | \(-4533623980032\) | \([2]\) | \(2048\) | \(1.1139\) | |
1344.g5 | 1344a2 | \([0, -1, 0, -257, -255]\) | \(7189057/3969\) | \(1040449536\) | \([2, 2]\) | \(512\) | \(0.42078\) | |
1344.g6 | 1344a1 | \([0, -1, 0, 63, -63]\) | \(103823/63\) | \(-16515072\) | \([2]\) | \(256\) | \(0.074205\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1344.g have rank \(1\).
Complex multiplication
The elliptic curves in class 1344.g do not have complex multiplication.Modular form 1344.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.