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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 317889p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
317889.p1 | 317889p1 | \([0, 0, 1, -41574, -3815640]\) | \(-2258403328/480491\) | \(-1690724708466651\) | \([]\) | \(1658880\) | \(1.6434\) | \(\Gamma_0(N)\)-optimal |
317889.p2 | 317889p2 | \([0, 0, 1, 293046, 22067217]\) | \(790939860992/517504691\) | \(-1820966402744482851\) | \([]\) | \(4976640\) | \(2.1927\) |
Rank
sage: E.rank()
The elliptic curves in class 317889p have rank \(1\).
Complex multiplication
The elliptic curves in class 317889p do not have complex multiplication.Modular form 317889.2.a.p
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.