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Properties

Label 1470.d
Number of curves $8$
Conductor $1470$
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, 0, -261342, 51314796]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -261342, 51314796]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -261342, 51314796]) 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 1470.d have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 1470.d do not have complex multiplication.

Modular form 1470.2.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);
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} - q^{12} - 2 q^{13} - q^{15} + q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)

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 1470.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
1470.d1 1470d8 \([1, 1, 0, -261342, 51314796]\) \(16778985534208729/81000\) \(9529569000\) \([2]\) \(6912\) \(1.5371\)  
1470.d2 1470d7 \([1, 1, 0, -22222, 164284]\) \(10316097499609/5859375000\) \(689349609375000\) \([2]\) \(6912\) \(1.5371\)  
1470.d3 1470d6 \([1, 1, 0, -16342, 795796]\) \(4102915888729/9000000\) \(1058841000000\) \([2, 2]\) \(3456\) \(1.1905\)  
1470.d4 1470d4 \([1, 1, 0, -14137, -652889]\) \(2656166199049/33750\) \(3970653750\) \([2]\) \(2304\) \(0.98781\)  
1470.d5 1470d5 \([1, 1, 0, -3357, 63099]\) \(35578826569/5314410\) \(625235022090\) \([2]\) \(2304\) \(0.98781\)  
1470.d6 1470d2 \([1, 1, 0, -907, -9911]\) \(702595369/72900\) \(8576612100\) \([2, 2]\) \(1152\) \(0.64124\)  
1470.d7 1470d3 \([1, 1, 0, -662, 21204]\) \(-273359449/1536000\) \(-180708864000\) \([2]\) \(1728\) \(0.84397\)  
1470.d8 1470d1 \([1, 1, 0, 73, -699]\) \(357911/2160\) \(-254121840\) \([2]\) \(576\) \(0.29467\) \(\Gamma_0(N)\)-optimal