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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 21675x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21675.h2 | 21675x1 | \([1, 0, 0, -873, -700488]\) | \(-24389/70227\) | \(-211888632270375\) | \([2]\) | \(92160\) | \(1.4279\) | \(\Gamma_0(N)\)-optimal |
21675.h1 | 21675x2 | \([1, 0, 0, -123698, -16544913]\) | \(69375867029/1003833\) | \(3028761037747125\) | \([2]\) | \(184320\) | \(1.7744\) |
Rank
sage: E.rank()
The elliptic curves in class 21675x have rank \(1\).
Complex multiplication
The elliptic curves in class 21675x do not have complex multiplication.Modular form 21675.2.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 Cremona 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 Cremona labels.