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Properties

Label 394944hk
Number of curves $4$
Conductor $394944$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("hk1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 394944hk have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(11\)\(1\)
\(17\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 394944hk do not have complex multiplication.

Modular form 394944.2.a.hk

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + 2 q^{7} + q^{9} - 4 q^{13} + q^{17} - 8 q^{19} + O(q^{20})\) 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 Cremona 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 sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.

Elliptic curves in class 394944hk

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
394944.hk3 394944hk1 \([0, 1, 0, -1590989473, -24179701612321]\) \(959024269496848362625/11151660319506432\) \(5178876626805754214574194688\) \([2]\) \(265420800\) \(4.1303\) \(\Gamma_0(N)\)-optimal
394944.hk4 394944hk2 \([0, 1, 0, -322212513, -61681449729825]\) \(-7966267523043306625/3534510366354604032\) \(-1641441058915491164594884313088\) \([2]\) \(530841600\) \(4.4769\)  
394944.hk1 394944hk3 \([0, 1, 0, -128508334753, -17731525731909409]\) \(505384091400037554067434625/815656731648\) \(378794319590211591340032\) \([2]\) \(796262400\) \(4.6796\)  
394944.hk2 394944hk4 \([0, 1, 0, -128507095713, -17731884750935841]\) \(-505369473241574671219626625/20303219722982711328\) \(-9428898214226683510900131889152\) \([2]\) \(1592524800\) \(5.0262\)