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Properties

Label 97104cb
Number of curves $6$
Conductor $97104$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 97104cb have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(7\)\(1 + T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + T + 5 T^{2}\) 1.5.b
\(11\) \( 1 - T + 11 T^{2}\) 1.11.ab
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 + 3 T + 23 T^{2}\) 1.23.d
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 97104cb do not have complex multiplication.

Modular form 97104.2.a.cb

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

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 97104cb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
97104.bf5 97104cb1 \([0, -1, 0, -324808352, -2250216630528]\) \(38331145780597164097/55468445663232\) \(5484025587789877915680768\) \([2]\) \(26542080\) \(3.6492\) \(\Gamma_0(N)\)-optimal
97104.bf4 97104cb2 \([0, -1, 0, -419507872, -831011744000]\) \(82582985847542515777/44772582831427584\) \(4426552555117767382234300416\) \([2, 2]\) \(53084160\) \(3.9957\)  
97104.bf6 97104cb3 \([0, -1, 0, 1618011488, -6537695967488]\) \(4738217997934888496063/2928751705237796928\) \(-289558308327608257567080579072\) \([4]\) \(106168320\) \(4.3423\)  
97104.bf2 97104cb4 \([0, -1, 0, -3972219552, 95703270024960]\) \(70108386184777836280897/552468975892674624\) \(54621216874368083277205733376\) \([2, 2]\) \(106168320\) \(4.3423\)  
97104.bf3 97104cb5 \([0, -1, 0, -1353000992, 220023955131648]\) \(-2770540998624539614657/209924951154647363208\) \(-20754751460626146918814356897792\) \([2]\) \(212336640\) \(4.6889\)  
97104.bf1 97104cb6 \([0, -1, 0, -63434824992, 6149519774744832]\) \(285531136548675601769470657/17941034271597192\) \(1773784894103723876787191808\) \([2]\) \(212336640\) \(4.6889\)