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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 175175be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 175175.bd4 | 175175be1 | \([1, -1, 0, -19217, 8650816]\) | \(-426957777/17320303\) | \(-31839317619484375\) | \([2]\) | \(933888\) | \(1.8468\) | \(\Gamma_0(N)\)-optimal |
| 175175.bd3 | 175175be2 | \([1, -1, 0, -760342, 253963191]\) | \(26444947540257/169338169\) | \(311288535073140625\) | \([2, 2]\) | \(1867776\) | \(2.1933\) | |
| 175175.bd1 | 175175be3 | \([1, -1, 0, -12146717, 16297365566]\) | \(107818231938348177/4463459\) | \(8205023248296875\) | \([2]\) | \(3735552\) | \(2.5399\) | |
| 175175.bd2 | 175175be4 | \([1, -1, 0, -1231967, -98340684]\) | \(112489728522417/62811265517\) | \(115463790262648953125\) | \([2]\) | \(3735552\) | \(2.5399\) |
Rank
sage: E.rank()
The elliptic curves in class 175175be have rank \(1\).
Complex multiplication
The elliptic curves in class 175175be do not have complex multiplication.Modular form 175175.2.a.be
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
