Properties

Label 19320bb
Number of curves $6$
Conductor $19320$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 19320bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
19320.ba6 19320bb1 \([0, 1, 0, 2305, 14010]\) \(84611246065664/53699121315\) \(-859185941040\) \([4]\) \(28672\) \(0.97893\) \(\Gamma_0(N)\)-optimal
19320.ba5 19320bb2 \([0, 1, 0, -9700, 105248]\) \(394315384276816/208332909225\) \(53333224761600\) \([2, 4]\) \(57344\) \(1.3255\)  
19320.ba3 19320bb3 \([0, 1, 0, -89080, -10182400]\) \(76343005935514084/694180580625\) \(710840914560000\) \([2, 2]\) \(114688\) \(1.6721\)  
19320.ba2 19320bb4 \([0, 1, 0, -122400, 16424208]\) \(198048499826486404/242568272835\) \(248389911383040\) \([4]\) \(114688\) \(1.6721\)  
19320.ba1 19320bb5 \([0, 1, 0, -1422160, -653260192]\) \(155324313723954725282/13018359375\) \(26661600000000\) \([2]\) \(229376\) \(2.0187\)  
19320.ba4 19320bb6 \([0, 1, 0, -26080, -24244000]\) \(-957928673903042/123339801817575\) \(-252599914122393600\) \([2]\) \(229376\) \(2.0187\)  

Rank

sage: E.rank()
 

The elliptic curves in class 19320bb have rank \(1\).

Complex multiplication

The elliptic curves in class 19320bb do not have complex multiplication.

Modular form 19320.2.a.bb

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{7} + q^{9} - 4q^{11} - 2q^{13} + q^{15} + 2q^{17} - 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.