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Properties

Label 1694f
Number of curves $6$
Conductor $1694$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -63, -395]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -63, -395]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -63, -395]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 1694f have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(7\)\(1 - T\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + T + 3 T^{2}\) 1.3.b
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(13\) \( 1 - 7 T + 13 T^{2}\) 1.13.ah
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 - T + 23 T^{2}\) 1.23.ab
\(29\) \( 1 + 8 T + 29 T^{2}\) 1.29.i
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 1694f do not have complex multiplication.

Modular form 1694.2.a.f

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} - q^{7} + q^{8} + q^{9} - 2 q^{12} + 4 q^{13} - q^{14} + q^{16} - 6 q^{17} + q^{18} - 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 1694f

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1694.e5 1694f1 \([1, 0, 0, -63, -395]\) \(-15625/28\) \(-49603708\) \([2]\) \(480\) \(0.16686\) \(\Gamma_0(N)\)-optimal
1694.e4 1694f2 \([1, 0, 0, -1273, -17577]\) \(128787625/98\) \(173612978\) \([2]\) \(960\) \(0.51344\)  
1694.e6 1694f3 \([1, 0, 0, 542, 8196]\) \(9938375/21952\) \(-38889307072\) \([2]\) \(1440\) \(0.71617\)  
1694.e3 1694f4 \([1, 0, 0, -4298, 88540]\) \(4956477625/941192\) \(1667379040712\) \([2]\) \(2880\) \(1.0627\)  
1694.e2 1694f5 \([1, 0, 0, -20633, 1142329]\) \(-548347731625/1835008\) \(-3250828607488\) \([2]\) \(4320\) \(1.2655\)  
1694.e1 1694f6 \([1, 0, 0, -330393, 73068601]\) \(2251439055699625/25088\) \(44444922368\) \([2]\) \(8640\) \(1.6120\)