Properties

Label 784.f
Number of curves $4$
Conductor $784$
CM \(\Q(\sqrt{-7}) \)
Rank $1$
Graph

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Copy content sage:E = EllipticCurve("f1") E.isogeny_class()
 

Elliptic curves in class 784.f

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
784.f1 784h4 \([0, 0, 0, -29155, -1915998]\) \(16581375\) \(165288374272\) \([2]\) \(896\) \(1.2135\)   \(-28\)
784.f2 784h3 \([0, 0, 0, -1715, -33614]\) \(-3375\) \(-165288374272\) \([2]\) \(448\) \(0.86696\)   \(-7\)
784.f3 784h2 \([0, 0, 0, -595, 5586]\) \(16581375\) \(1404928\) \([2]\) \(128\) \(0.24058\)   \(-28\)
784.f4 784h1 \([0, 0, 0, -35, 98]\) \(-3375\) \(-1404928\) \([2]\) \(64\) \(-0.10599\) \(\Gamma_0(N)\)-optimal \(-7\)

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 784.f have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 3 T^{2}\) 1.3.a
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 13 T^{2}\) 1.13.a
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

Each elliptic curve in class 784.f has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).

Modular form 784.2.a.f

Copy content sage:E.q_eigenform(10)
 
\(q - 3 q^{9} - 4 q^{11} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.