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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 137904.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
137904.p1 | 137904bi2 | \([0, -1, 0, -888, 7344]\) | \(8615125/2601\) | \(23406170112\) | \([2]\) | \(98304\) | \(0.69485\) | |
137904.p2 | 137904bi1 | \([0, -1, 0, 152, 688]\) | \(42875/51\) | \(-458944512\) | \([2]\) | \(49152\) | \(0.34827\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 137904.p have rank \(2\).
Complex multiplication
The elliptic curves in class 137904.p do not have complex multiplication.Modular form 137904.2.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 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.