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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 250025.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 250025.d1 | 250025d4 | \([1, -1, 0, -833416667, -9260439837384]\) | \(4097228930565632081600480001/6250625\) | \(97666015625\) | \([2]\) | \(21676032\) | \(3.2492\) | |
| 250025.d2 | 250025d2 | \([1, -1, 0, -52088542, -144684603009]\) | \(1000300060007000600030001/39070312890625\) | \(610473638916015625\) | \([2, 2]\) | \(10838016\) | \(2.9027\) | |
| 250025.d3 | 250025d3 | \([1, -1, 0, -52010417, -145140306134]\) | \(-995805907172059099580001/6252500375025000625\) | \(-97695318359765634765625\) | \([2]\) | \(21676032\) | \(3.2492\) | |
| 250025.d4 | 250025d1 | \([1, -1, 0, -3260417, -2252962384]\) | \(245314493562818780001/1526031494140625\) | \(23844242095947265625\) | \([2]\) | \(5419008\) | \(2.5561\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 250025.d have rank \(1\).
Complex multiplication
The elliptic curves in class 250025.d do not have complex multiplication.Modular form 250025.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
