Properties

Label 48510.cu
Number of curves $8$
Conductor $48510$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cu1")
 
E.isogeny_class()
 

Elliptic curves in class 48510.cu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
48510.cu1 48510dk8 \([1, -1, 1, -11231382938, -458135277511719]\) \(1826870018430810435423307849/7641104625000000000\) \(655347903841409625000000000\) \([2]\) \(63700992\) \(4.3545\)  
48510.cu2 48510dk6 \([1, -1, 1, -712862618, -6924415048743]\) \(467116778179943012100169/28800309694464000000\) \(2470090846092872454144000000\) \([2, 2]\) \(31850496\) \(4.0079\)  
48510.cu3 48510dk5 \([1, -1, 1, -193058123, -90611800653]\) \(9278380528613437145689/5328033205714065000\) \(456964740613290390191865000\) \([2]\) \(21233664\) \(3.8052\)  
48510.cu4 48510dk3 \([1, -1, 1, -134835098, 469250565081]\) \(3160944030998056790089/720291785342976000\) \(61776632417031706116096000\) \([2]\) \(15925248\) \(3.6614\)  
48510.cu5 48510dk2 \([1, -1, 1, -126502403, 545474526531]\) \(2610383204210122997209/12104550027662400\) \(1038160302323046745550400\) \([2, 2]\) \(10616832\) \(3.4586\)  
48510.cu6 48510dk1 \([1, -1, 1, -126361283, 546756855747]\) \(2601656892010848045529/56330588160\) \(4831256040131727360\) \([2]\) \(5308416\) \(3.1120\) \(\Gamma_0(N)\)-optimal
48510.cu7 48510dk4 \([1, -1, 1, -62204603, 1099490090451]\) \(-310366976336070130009/5909282337130963560\) \(-506816223949537013533550760\) \([2]\) \(21233664\) \(3.8052\)  
48510.cu8 48510dk7 \([1, -1, 1, 557217382, -28915088200743]\) \(223090928422700449019831/4340371122724101696000\) \(-372256794896461155675441216000\) \([2]\) \(63700992\) \(4.3545\)  

Rank

sage: E.rank()
 

The elliptic curves in class 48510.cu have rank \(1\).

Complex multiplication

The elliptic curves in class 48510.cu do not have complex multiplication.

Modular form 48510.2.a.cu

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} + q^{11} - 2 q^{13} + q^{16} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 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.