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Properties

Label 31878bl
Number of curves $4$
Conductor $31878$
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, -1, 1, 330160, 297908003]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, 330160, 297908003]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, 330160, 297908003]) 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 31878bl have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(7\)\(1 - T\)
\(11\)\(1 + T\)
\(23\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - T + 5 T^{2}\) 1.5.ab
\(13\) \( 1 - 5 T + 13 T^{2}\) 1.13.af
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(29\) \( 1 - 3 T + 29 T^{2}\) 1.29.ad
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 31878bl do not have complex multiplication.

Modular form 31878.2.a.bl

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);
 
\(q + q^{2} + q^{4} + q^{7} + q^{8} - q^{11} - 4 q^{13} + q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

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 31878bl

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
31878.bg4 31878bl1 \([1, -1, 1, 330160, 297908003]\) \(5459725204437026375/55780815891710448\) \(-40664214785056916592\) \([2]\) \(958464\) \(2.4438\) \(\Gamma_0(N)\)-optimal
31878.bg3 31878bl2 \([1, -1, 1, -5180180, 4212453539]\) \(21087770069125509765625/1694619018457399188\) \(1235377264455444008052\) \([2]\) \(1916928\) \(2.7903\)  
31878.bg2 31878bl3 \([1, -1, 1, -25691495, 50163161375]\) \(-2572552807198813678947625/2038409681283182592\) \(-1486000657655440109568\) \([6]\) \(2875392\) \(2.9931\)  
31878.bg1 31878bl4 \([1, -1, 1, -411142055, 3208853410463]\) \(10543186518294206197228515625/6611719873695552\) \(4819943787924057408\) \([6]\) \(5750784\) \(3.3396\)