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SageMath
E = EllipticCurve("gh1")
E.isogeny_class()
Elliptic curves in class 369600.gh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
369600.gh1 | 369600gh6 | \([0, -1, 0, -7230433, -7480911263]\) | \(10206027697760497/5557167\) | \(22762156032000000\) | \([2]\) | \(10485760\) | \(2.4672\) | |
369600.gh2 | 369600gh4 | \([0, -1, 0, -454433, -115399263]\) | \(2533811507137/58110129\) | \(238019088384000000\) | \([2, 2]\) | \(5242880\) | \(2.1207\) | |
369600.gh3 | 369600gh2 | \([0, -1, 0, -62433, 3376737]\) | \(6570725617/2614689\) | \(10709766144000000\) | \([2, 2]\) | \(2621440\) | \(1.7741\) | |
369600.gh4 | 369600gh1 | \([0, -1, 0, -54433, 4904737]\) | \(4354703137/1617\) | \(6623232000000\) | \([2]\) | \(1310720\) | \(1.4275\) | \(\Gamma_0(N)\)-optimal |
369600.gh5 | 369600gh5 | \([0, -1, 0, 49567, -357823263]\) | \(3288008303/13504609503\) | \(-55314880524288000000\) | \([2]\) | \(10485760\) | \(2.4672\) | |
369600.gh6 | 369600gh3 | \([0, -1, 0, 201567, 24232737]\) | \(221115865823/190238433\) | \(-779216621568000000\) | \([2]\) | \(5242880\) | \(2.1207\) |
Rank
sage: E.rank()
The elliptic curves in class 369600.gh have rank \(0\).
Complex multiplication
The elliptic curves in class 369600.gh do not have complex multiplication.Modular form 369600.2.a.gh
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.