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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 36414bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
36414.h1 | 36414bn1 | \([1, -1, 0, -18261, -945459]\) | \(-11060825617/2744\) | \(-167073403896\) | \([]\) | \(90720\) | \(1.1410\) | \(\Gamma_0(N)\)-optimal |
36414.h2 | 36414bn2 | \([1, -1, 0, 7749, -3364389]\) | \(845095823/80707214\) | \(-4914004723740126\) | \([3]\) | \(272160\) | \(1.6903\) |
Rank
sage: E.rank()
The elliptic curves in class 36414bn have rank \(0\).
Complex multiplication
The elliptic curves in class 36414bn do not have complex multiplication.Modular form 36414.2.a.bn
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.