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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 6370.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6370.h1 | 6370c3 | \([1, 1, 0, -10168, 374592]\) | \(988345570681/44994560\) | \(5293564989440\) | \([2]\) | \(20736\) | \(1.2034\) | |
| 6370.h2 | 6370c1 | \([1, 1, 0, -1593, -25003]\) | \(3803721481/26000\) | \(3058874000\) | \([2]\) | \(6912\) | \(0.65410\) | \(\Gamma_0(N)\)-optimal |
| 6370.h3 | 6370c2 | \([1, 1, 0, -613, -54207]\) | \(-217081801/10562500\) | \(-1242667562500\) | \([2]\) | \(13824\) | \(1.0007\) | |
| 6370.h4 | 6370c4 | \([1, 1, 0, 5512, 1443968]\) | \(157376536199/7722894400\) | \(-908590803265600\) | \([2]\) | \(41472\) | \(1.5500\) |
Rank
sage: E.rank()
The elliptic curves in class 6370.h have rank \(0\).
Complex multiplication
The elliptic curves in class 6370.h do not have complex multiplication.Modular form 6370.2.a.h
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
