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Properties

Label 64350dj
Number of curves $6$
Conductor $64350$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dj1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 64350dj have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1\)
\(11\)\(1 + T\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + T + 7 T^{2}\) 1.7.b
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - T + 19 T^{2}\) 1.19.ab
\(23\) \( 1 + 2 T + 23 T^{2}\) 1.23.c
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 64350dj do not have complex multiplication.

Modular form 64350.2.a.dj

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 64350dj

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
64350.dr5 64350dj1 \([1, -1, 1, -639005, -85527003]\) \(2533309721804161/1187575234560\) \(13527224156160000000\) \([2]\) \(1474560\) \(2.3653\) \(\Gamma_0(N)\)-optimal
64350.dr4 64350dj2 \([1, -1, 1, -5247005, 4568552997]\) \(1402524686897642881/20523074457600\) \(233770644993600000000\) \([2, 2]\) \(2949120\) \(2.7118\)  
64350.dr6 64350dj3 \([1, -1, 1, -567005, 12421592997]\) \(-1769848555063681/5850659851882560\) \(-66642672375349785000000\) \([2]\) \(5898240\) \(3.0584\)  
64350.dr2 64350dj4 \([1, -1, 1, -83655005, 294521336997]\) \(5683972151443376419201/1244117160000\) \(14171272025625000000\) \([2, 2]\) \(5898240\) \(3.0584\)  
64350.dr3 64350dj5 \([1, -1, 1, -83358005, 296716166997]\) \(-5623647484692626737921/84122230603125000\) \(-958204782963720703125000\) \([2]\) \(11796480\) \(3.4050\)  
64350.dr1 64350dj6 \([1, -1, 1, -1338480005, 18848363786997]\) \(23281546263261052473907201/1115400\) \(12705103125000\) \([2]\) \(11796480\) \(3.4050\)