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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 1344.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1344.s1 | 1344r5 | \([0, 1, 0, -50177, 4309503]\) | \(53297461115137/147\) | \(38535168\) | \([2]\) | \(2048\) | \(1.1139\) | |
1344.s2 | 1344r4 | \([0, 1, 0, -3137, 66495]\) | \(13027640977/21609\) | \(5664669696\) | \([2, 2]\) | \(1024\) | \(0.76735\) | |
1344.s3 | 1344r3 | \([0, 1, 0, -2497, -48577]\) | \(6570725617/45927\) | \(12039487488\) | \([2]\) | \(1024\) | \(0.76735\) | |
1344.s4 | 1344r6 | \([0, 1, 0, -2177, 108927]\) | \(-4354703137/17294403\) | \(-4533623980032\) | \([2]\) | \(2048\) | \(1.1139\) | |
1344.s5 | 1344r2 | \([0, 1, 0, -257, 255]\) | \(7189057/3969\) | \(1040449536\) | \([2, 2]\) | \(512\) | \(0.42078\) | |
1344.s6 | 1344r1 | \([0, 1, 0, 63, 63]\) | \(103823/63\) | \(-16515072\) | \([2]\) | \(256\) | \(0.074205\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1344.s have rank \(0\).
Complex multiplication
The elliptic curves in class 1344.s do not have complex multiplication.Modular form 1344.2.a.s
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.