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SageMath
E = EllipticCurve("ge1")
E.isogeny_class()
Elliptic curves in class 80850.ge
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
80850.ge1 | 80850fx8 | \([1, 0, 0, -31198285938, 2121007054575492]\) | \(1826870018430810435423307849/7641104625000000000\) | \(14046379969166015625000000000\) | \([2]\) | \(191102976\) | \(4.6099\) | |
80850.ge2 | 80850fx6 | \([1, 0, 0, -1980173938, 32058137135492]\) | \(467116778179943012100169/28800309694464000000\) | \(52942619300687424000000000000\) | \([2, 2]\) | \(95551488\) | \(4.2633\) | |
80850.ge3 | 80850fx5 | \([1, 0, 0, -536272563, 419677834617]\) | \(9278380528613437145689/5328033205714065000\) | \(9794340290922719268515625000\) | \([2]\) | \(63700992\) | \(4.0606\) | |
80850.ge4 | 80850fx3 | \([1, 0, 0, -374541938, -2172331472508]\) | \(3160944030998056790089/720291785342976000\) | \(1324087628965871616000000000\) | \([2]\) | \(47775744\) | \(3.9168\) | |
80850.ge5 | 80850fx2 | \([1, 0, 0, -351395563, -2525227898383]\) | \(2610383204210122997209/12104550027662400\) | \(22251378221944589025000000\) | \([2, 2]\) | \(31850496\) | \(3.7140\) | |
80850.ge6 | 80850fx1 | \([1, 0, 0, -351003563, -2531164738383]\) | \(2601656892010848045529/56330588160\) | \(103550583850560000000\) | \([2]\) | \(15925248\) | \(3.3675\) | \(\Gamma_0(N)\)-optimal |
80850.ge7 | 80850fx4 | \([1, 0, 0, -172790563, -5090174303383]\) | \(-310366976336070130009/5909282337130963560\) | \(-10862830588767511435475625000\) | \([2]\) | \(63700992\) | \(4.0606\) | |
80850.ge8 | 80850fx7 | \([1, 0, 0, 1547826062, 133865633135492]\) | \(223090928422700449019831/4340371122724101696000\) | \(-7978755034646372506761000000000\) | \([2]\) | \(191102976\) | \(4.6099\) |
Rank
sage: E.rank()
The elliptic curves in class 80850.ge have rank \(0\).
Complex multiplication
The elliptic curves in class 80850.ge do not have complex multiplication.Modular form 80850.2.a.ge
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.