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Properties

Label 6762.h
Number of curves $4$
Conductor $6762$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("h1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 6762.h have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(7\)\(1\)
\(23\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 - 6 T + 11 T^{2}\) 1.11.ag
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 6762.h do not have complex multiplication.

Modular form 6762.2.a.h

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 6762.h

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6762.h1 6762f4 \([1, 1, 0, -74308035, 245321703117]\) \(385693937170561837203625/2159357734550274048\) \(254046278112105191473152\) \([2]\) \(1382400\) \(3.3333\)  
6762.h2 6762f2 \([1, 1, 0, -5487780, -4716747216]\) \(155355156733986861625/8291568305839392\) \(975494719613698629408\) \([2]\) \(460800\) \(2.7840\)  
6762.h3 6762f3 \([1, 1, 0, -2054595, 8084758221]\) \(-8152944444844179625/235342826399858688\) \(-27687848183116974784512\) \([2]\) \(691200\) \(2.9867\)  
6762.h4 6762f1 \([1, 1, 0, 227580, -294201648]\) \(11079872671250375/324440155855872\) \(-38170059896287484928\) \([2]\) \(230400\) \(2.4374\) \(\Gamma_0(N)\)-optimal