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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 34914e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34914.i3 | 34914e1 | \([1, 1, 0, -23324414, 43347235188]\) | \(9479576797126950457/138431496192\) | \(20492829604382834688\) | \([2]\) | \(3041280\) | \(2.8438\) | \(\Gamma_0(N)\)-optimal |
34914.i2 | 34914e2 | \([1, 1, 0, -24001534, 40696039540]\) | \(10329367348538068537/1142220445749504\) | \(169089619120504095949056\) | \([2, 2]\) | \(6082560\) | \(3.1903\) | |
34914.i4 | 34914e3 | \([1, 1, 0, 32326386, 203202088740]\) | \(25236442759706220743/135084570146078256\) | \(-19997364431757554490529584\) | \([2]\) | \(12165120\) | \(3.5369\) | |
34914.i1 | 34914e4 | \([1, 1, 0, -91163374, -291472988732]\) | \(566001880654007645497/79690973341699632\) | \(11797124083913805794092848\) | \([2]\) | \(12165120\) | \(3.5369\) |
Rank
sage: E.rank()
The elliptic curves in class 34914e have rank \(0\).
Complex multiplication
The elliptic curves in class 34914e do not have complex multiplication.Modular form 34914.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.