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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 22050x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22050.o2 | 22050x1 | \([1, -1, 0, 181683, -39333659]\) | \(10100279/16000\) | \(-1050634982250000000\) | \([]\) | \(338688\) | \(2.1432\) | \(\Gamma_0(N)\)-optimal |
22050.o1 | 22050x2 | \([1, -1, 0, -1747692, 1544683216]\) | \(-8990558521/10485760\) | \(-688544141967360000000\) | \([]\) | \(1016064\) | \(2.6925\) |
Rank
sage: E.rank()
The elliptic curves in class 22050x have rank \(0\).
Complex multiplication
The elliptic curves in class 22050x do not have complex multiplication.Modular form 22050.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.