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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 21450.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21450.x1 | 21450bd2 | \([1, 0, 1, -403318526, -3117633430552]\) | \(464352938845529653759213009/2445173327025000\) | \(38205833234765625000\) | \([2]\) | \(3870720\) | \(3.3742\) | |
21450.x2 | 21450bd1 | \([1, 0, 1, -25193526, -48770930552]\) | \(-113180217375258301213009/260161419375000000\) | \(-4065022177734375000000\) | \([2]\) | \(1935360\) | \(3.0276\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 21450.x have rank \(0\).
Complex multiplication
The elliptic curves in class 21450.x do not have complex multiplication.Modular form 21450.2.a.x
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.