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Properties

Label 388542.dh
Number of curves $4$
Conductor $388542$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dh1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 388542.dh have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(7\)\(1 + T\)
\(11\)\(1 + T\)
\(29\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 388542.dh do not have complex multiplication.

Modular form 388542.2.a.dh

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + 2 q^{10} - q^{11} + q^{12} + 2 q^{13} - q^{14} + 2 q^{15} + q^{16} - 6 q^{17} + q^{18} + 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 388542.dh

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
388542.dh1 388542dh3 \([1, 0, 0, -77375802, 261966121092]\) \(86129359107301290313/9166294368\) \(5452325657237356128\) \([2]\) \(48168960\) \(3.0230\)  
388542.dh2 388542dh2 \([1, 0, 0, -4847962, 4071627620]\) \(21184262604460873/216872764416\) \(129000977964375745536\) \([2, 2]\) \(24084480\) \(2.6764\)  
388542.dh3 388542dh4 \([1, 0, 0, -1214842, 10033577540]\) \(-333345918055753/72923718045024\) \(-43376728147208803204704\) \([2]\) \(48168960\) \(3.0230\)  
388542.dh4 388542dh1 \([1, 0, 0, -542042, -50860188]\) \(29609739866953/15259926528\) \(9076960175600959488\) \([2]\) \(12042240\) \(2.3298\) \(\Gamma_0(N)\)-optimal