Properties

Label 277200ib
Number of curves $8$
Conductor $277200$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("ib1")
 
E.isogeny_class()
 

Elliptic curves in class 277200ib

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
277200.ib6 277200ib1 \([0, 0, 0, -1031520675, 12751609689250]\) \(2601656892010848045529/56330588160\) \(2628159921192960000000\) \([2]\) \(63700992\) \(3.6370\) \(\Gamma_0(N)\)-optimal
277200.ib5 277200ib2 \([0, 0, 0, -1032672675, 12721700313250]\) \(2610383204210122997209/12104550027662400\) \(564749886090616934400000000\) \([2, 2]\) \(127401984\) \(3.9835\)  
277200.ib4 277200ib3 \([0, 0, 0, -1100694675, 10943833383250]\) \(3160944030998056790089/720291785342976000\) \(33605933536961888256000000000\) \([2]\) \(191102976\) \(4.1863\)  
277200.ib7 277200ib4 \([0, 0, 0, -507792675, 25643721033250]\) \(-310366976336070130009/5909282337130963560\) \(-275703476721182235855360000000\) \([2]\) \(254803968\) \(4.3301\)  
277200.ib3 277200ib5 \([0, 0, 0, -1575984675, -2114520470750]\) \(9278380528613437145689/5328033205714065000\) \(248584717245795416640000000000\) \([2]\) \(254803968\) \(4.3301\)  
277200.ib2 277200ib6 \([0, 0, 0, -5819286675, -161506548440750]\) \(467116778179943012100169/28800309694464000000\) \(1343707249104912384000000000000\) \([2, 2]\) \(382205952\) \(4.5328\)  
277200.ib8 277200ib7 \([0, 0, 0, 4548713325, -674401140440750]\) \(223090928422700449019831/4340371122724101696000\) \(-202504355101815688728576000000000\) \([2]\) \(764411904\) \(4.8794\)  
277200.ib1 277200ib8 \([0, 0, 0, -91684758675, -10685436393176750]\) \(1826870018430810435423307849/7641104625000000000\) \(356503377384000000000000000000\) \([2]\) \(764411904\) \(4.8794\)  

Rank

sage: E.rank()
 

The elliptic curves in class 277200ib have rank \(0\).

Complex multiplication

The elliptic curves in class 277200ib do not have complex multiplication.

Modular form 277200.2.a.ib

sage: E.q_eigenform(10)
 
\(q + q^{7} - q^{11} - 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.