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SageMath
E = EllipticCurve("ib1")
E.isogeny_class()
Elliptic curves in class 277200.ib
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
277200.ib1 | 277200ib8 | \([0, 0, 0, -91684758675, -10685436393176750]\) | \(1826870018430810435423307849/7641104625000000000\) | \(356503377384000000000000000000\) | \([2]\) | \(764411904\) | \(4.8794\) | |
277200.ib2 | 277200ib6 | \([0, 0, 0, -5819286675, -161506548440750]\) | \(467116778179943012100169/28800309694464000000\) | \(1343707249104912384000000000000\) | \([2, 2]\) | \(382205952\) | \(4.5328\) | |
277200.ib3 | 277200ib5 | \([0, 0, 0, -1575984675, -2114520470750]\) | \(9278380528613437145689/5328033205714065000\) | \(248584717245795416640000000000\) | \([2]\) | \(254803968\) | \(4.3301\) | |
277200.ib4 | 277200ib3 | \([0, 0, 0, -1100694675, 10943833383250]\) | \(3160944030998056790089/720291785342976000\) | \(33605933536961888256000000000\) | \([2]\) | \(191102976\) | \(4.1863\) | |
277200.ib5 | 277200ib2 | \([0, 0, 0, -1032672675, 12721700313250]\) | \(2610383204210122997209/12104550027662400\) | \(564749886090616934400000000\) | \([2, 2]\) | \(127401984\) | \(3.9835\) | |
277200.ib6 | 277200ib1 | \([0, 0, 0, -1031520675, 12751609689250]\) | \(2601656892010848045529/56330588160\) | \(2628159921192960000000\) | \([2]\) | \(63700992\) | \(3.6370\) | \(\Gamma_0(N)\)-optimal |
277200.ib7 | 277200ib4 | \([0, 0, 0, -507792675, 25643721033250]\) | \(-310366976336070130009/5909282337130963560\) | \(-275703476721182235855360000000\) | \([2]\) | \(254803968\) | \(4.3301\) | |
277200.ib8 | 277200ib7 | \([0, 0, 0, 4548713325, -674401140440750]\) | \(223090928422700449019831/4340371122724101696000\) | \(-202504355101815688728576000000000\) | \([2]\) | \(764411904\) | \(4.8794\) |
Rank
sage: E.rank()
The elliptic curves in class 277200.ib have rank \(0\).
Complex multiplication
The elliptic curves in class 277200.ib do not have complex multiplication.Modular form 277200.2.a.ib
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.