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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 1008.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1008.h1 | 1008i6 | \([0, 0, 0, -393195, -94898662]\) | \(2251439055699625/25088\) | \(74912366592\) | \([2]\) | \(3456\) | \(1.6556\) | |
| 1008.h2 | 1008i5 | \([0, 0, 0, -24555, -1485286]\) | \(-548347731625/1835008\) | \(-5479304527872\) | \([2]\) | \(1728\) | \(1.3090\) | |
| 1008.h3 | 1008i4 | \([0, 0, 0, -5115, -115414]\) | \(4956477625/941192\) | \(2810384252928\) | \([2]\) | \(1152\) | \(1.1062\) | |
| 1008.h4 | 1008i2 | \([0, 0, 0, -1515, 22682]\) | \(128787625/98\) | \(292626432\) | \([2]\) | \(384\) | \(0.55694\) | |
| 1008.h5 | 1008i1 | \([0, 0, 0, -75, 506]\) | \(-15625/28\) | \(-83607552\) | \([2]\) | \(192\) | \(0.21037\) | \(\Gamma_0(N)\)-optimal |
| 1008.h6 | 1008i3 | \([0, 0, 0, 645, -10582]\) | \(9938375/21952\) | \(-65548320768\) | \([2]\) | \(576\) | \(0.75968\) |
Rank
sage: E.rank()
The elliptic curves in class 1008.h have rank \(1\).
Complex multiplication
The elliptic curves in class 1008.h do not have complex multiplication.Modular form 1008.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
