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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 162450.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162450.a1 | 162450da3 | \([1, -1, 0, -40001689692, 3079406547008716]\) | \(13209596798923694545921/92340\) | \(49483342796447812500\) | \([2]\) | \(265420800\) | \(4.3137\) | |
| 162450.a2 | 162450da4 | \([1, -1, 0, -2530972692, 46867159493716]\) | \(3345930611358906241/165622259047500\) | \(88753985479418698989492187500\) | \([2]\) | \(265420800\) | \(4.3137\) | |
| 162450.a3 | 162450da2 | \([1, -1, 0, -2500107192, 48116131951216]\) | \(3225005357698077121/8526675600\) | \(4569291873823991006250000\) | \([2, 2]\) | \(132710400\) | \(3.9672\) | |
| 162450.a4 | 162450da1 | \([1, -1, 0, -154329192, 771294577216]\) | \(-758575480593601/40535043840\) | \(-21721999887413474940000000\) | \([2]\) | \(66355200\) | \(3.6206\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 162450.a have rank \(1\).
Complex multiplication
The elliptic curves in class 162450.a do not have complex multiplication.Modular form 162450.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
