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Properties

Label 370.d
Number of curves $4$
Conductor $370$
CM no
Rank $0$
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, -5275, -147903]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -5275, -147903]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -5275, -147903]) 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 370.d have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(5\)\(1 - T\)
\(37\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 2 T + 3 T^{2}\) 1.3.c
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 370.d do not have complex multiplication.

Modular form 370.2.a.d

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);
 
\(q + q^{2} - 2 q^{3} + q^{4} + q^{5} - 2 q^{6} + 2 q^{7} + q^{8} + q^{9} + q^{10} - 2 q^{12} + 2 q^{13} + 2 q^{14} - 2 q^{15} + q^{16} + 6 q^{17} + q^{18} + 2 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 370.d

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
370.d1 370d3 \([1, 0, 0, -5275, -147903]\) \(16232905099479601/4052240\) \(4052240\) \([2]\) \(288\) \(0.64351\)  
370.d2 370d4 \([1, 0, 0, -5255, -149075]\) \(-16048965315233521/256572640900\) \(-256572640900\) \([2]\) \(576\) \(0.99008\)  
370.d3 370d1 \([1, 0, 0, -75, -143]\) \(46694890801/18944000\) \(18944000\) \([6]\) \(96\) \(0.094202\) \(\Gamma_0(N)\)-optimal
370.d4 370d2 \([1, 0, 0, 245, -975]\) \(1625964918479/1369000000\) \(-1369000000\) \([6]\) \(192\) \(0.44078\)