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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 43245.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 43245.h1 | 43245b8 | \([1, -1, 0, -18682020, -31075534089]\) | \(1114544804970241/405\) | \(262031024296845\) | \([2]\) | \(983040\) | \(2.5572\) | |
| 43245.h2 | 43245b6 | \([1, -1, 0, -1167795, -485188704]\) | \(272223782641/164025\) | \(106122564840222225\) | \([2, 2]\) | \(491520\) | \(2.2106\) | |
| 43245.h3 | 43245b7 | \([1, -1, 0, -951570, -670580019]\) | \(-147281603041/215233605\) | \(-139254029583339603645\) | \([2]\) | \(983040\) | \(2.5572\) | |
| 43245.h4 | 43245b4 | \([1, -1, 0, -692100, 221789205]\) | \(56667352321/15\) | \(9704852751735\) | \([2]\) | \(245760\) | \(1.8640\) | |
| 43245.h5 | 43245b3 | \([1, -1, 0, -86670, -4520529]\) | \(111284641/50625\) | \(32753878037105625\) | \([2, 2]\) | \(245760\) | \(1.8640\) | |
| 43245.h6 | 43245b2 | \([1, -1, 0, -43425, 3445200]\) | \(13997521/225\) | \(145572791276025\) | \([2, 2]\) | \(122880\) | \(1.5174\) | |
| 43245.h7 | 43245b1 | \([1, -1, 0, -180, 149931]\) | \(-1/15\) | \(-9704852751735\) | \([2]\) | \(61440\) | \(1.1709\) | \(\Gamma_0(N)\)-optimal |
| 43245.h8 | 43245b5 | \([1, -1, 0, 302535, -34177950]\) | \(4733169839/3515625\) | \(-2274574863687890625\) | \([2]\) | \(491520\) | \(2.2106\) |
Rank
sage: E.rank()
The elliptic curves in class 43245.h have rank \(1\).
Complex multiplication
The elliptic curves in class 43245.h do not have complex multiplication.Modular form 43245.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
