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Properties

Label 216384.fi
Number of curves $6$
Conductor $216384$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 0, -248164289, -1504792275393]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -248164289, -1504792275393]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -248164289, -1504792275393]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 216384.fi have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(7\)\(1\)
\(23\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 216384.fi do not have complex multiplication.

Modular form 216384.2.a.fi

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q + q^{3} - 2 q^{5} + q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 216384.fi

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
216384.fi1 216384l6 \([0, 1, 0, -248164289, -1504792275393]\) \(54804145548726848737/637608031452\) \(19664456198991733850112\) \([2]\) \(37748736\) \(3.4286\)  
216384.fi2 216384l3 \([0, 1, 0, -55551169, 159340362047]\) \(614716917569296417/19093020912\) \(588847465699970383872\) \([2]\) \(18874368\) \(3.0821\)  
216384.fi3 216384l4 \([0, 1, 0, -15912129, -22233837249]\) \(14447092394873377/1439452851984\) \(44394135835919152840704\) \([2, 2]\) \(18874368\) \(3.0821\)  
216384.fi4 216384l2 \([0, 1, 0, -3619009, 2266350911]\) \(169967019783457/26337394944\) \(812271056392462270464\) \([2, 2]\) \(9437184\) \(2.7355\)  
216384.fi5 216384l1 \([0, 1, 0, 395071, 195888447]\) \(221115865823/664731648\) \(-20500975099721023488\) \([2]\) \(4718592\) \(2.3889\) \(\Gamma_0(N)\)-optimal
216384.fi6 216384l5 \([0, 1, 0, 19650111, -107490751425]\) \(27207619911317663/177609314617308\) \(-5477645223306636530024448\) \([2]\) \(37748736\) \(3.4286\)