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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 194271d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
194271.i3 | 194271d1 | \([1, 0, 0, -129952, -14661025]\) | \(408023180713/80247321\) | \(47732977978573041\) | \([2]\) | \(1290240\) | \(1.9169\) | \(\Gamma_0(N)\)-optimal |
194271.i2 | 194271d2 | \([1, 0, 0, -638757, 183264120]\) | \(48455467135993/3635004681\) | \(2162185556202965601\) | \([2, 2]\) | \(2580480\) | \(2.2635\) | |
194271.i1 | 194271d3 | \([1, 0, 0, -10028522, 12222820803]\) | \(187519537050946633/1186707753\) | \(705881446695907713\) | \([2]\) | \(5160960\) | \(2.6101\) | |
194271.i4 | 194271d4 | \([1, 0, 0, 610128, 812951937]\) | \(42227808999767/504359959257\) | \(-300005065944673432497\) | \([2]\) | \(5160960\) | \(2.6101\) |
Rank
sage: E.rank()
The elliptic curves in class 194271d have rank \(1\).
Complex multiplication
The elliptic curves in class 194271d do not have complex multiplication.Modular form 194271.2.a.d
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.