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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 336.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
336.a1 | 336e5 | \([0, -1, 0, -12544, 544960]\) | \(53297461115137/147\) | \(602112\) | \([4]\) | \(256\) | \(0.76735\) | |
336.a2 | 336e4 | \([0, -1, 0, -784, 8704]\) | \(13027640977/21609\) | \(88510464\) | \([2, 4]\) | \(128\) | \(0.42078\) | |
336.a3 | 336e3 | \([0, -1, 0, -624, -5760]\) | \(6570725617/45927\) | \(188116992\) | \([2]\) | \(128\) | \(0.42078\) | |
336.a4 | 336e6 | \([0, -1, 0, -544, 13888]\) | \(-4354703137/17294403\) | \(-70837874688\) | \([4]\) | \(256\) | \(0.76735\) | |
336.a5 | 336e2 | \([0, -1, 0, -64, 64]\) | \(7189057/3969\) | \(16257024\) | \([2, 2]\) | \(64\) | \(0.074205\) | |
336.a6 | 336e1 | \([0, -1, 0, 16, 0]\) | \(103823/63\) | \(-258048\) | \([2]\) | \(32\) | \(-0.27237\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 336.a have rank \(1\).
Complex multiplication
The elliptic curves in class 336.a do not have complex multiplication.Modular form 336.2.a.a
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.