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Properties

Label 14415.d
Number of curves $8$
Conductor $14415$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -2075780, 1150945707]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -2075780, 1150945707]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -2075780, 1150945707]) 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 14415.d have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(31\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 14415.d do not have complex multiplication.

Modular form 14415.2.a.d

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);
 
\(q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} + 3 q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} + 2 q^{13} + q^{15} - q^{16} - 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

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 14415.d

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
14415.d1 14415g7 \([1, 0, 0, -2075780, 1150945707]\) \(1114544804970241/405\) \(359438990805\) \([2]\) \(122880\) \(2.0079\)  
14415.d2 14415g5 \([1, 0, 0, -129755, 17969952]\) \(272223782641/164025\) \(145572791276025\) \([2, 2]\) \(61440\) \(1.6613\)  
14415.d3 14415g8 \([1, 0, 0, -105730, 24836297]\) \(-147281603041/215233605\) \(-191020616712400005\) \([2]\) \(122880\) \(2.0079\)  
14415.d4 14415g3 \([1, 0, 0, -76900, -8214415]\) \(56667352321/15\) \(13312555215\) \([2]\) \(30720\) \(1.3147\)  
14415.d5 14415g4 \([1, 0, 0, -9630, 167427]\) \(111284641/50625\) \(44929873850625\) \([2, 2]\) \(30720\) \(1.3147\)  
14415.d6 14415g2 \([1, 0, 0, -4825, -127600]\) \(13997521/225\) \(199688328225\) \([2, 2]\) \(15360\) \(0.96814\)  
14415.d7 14415g1 \([1, 0, 0, -20, -5553]\) \(-1/15\) \(-13312555215\) \([2]\) \(7680\) \(0.62157\) \(\Gamma_0(N)\)-optimal
14415.d8 14415g6 \([1, 0, 0, 33615, 1265850]\) \(4733169839/3515625\) \(-3120130128515625\) \([2]\) \(61440\) \(1.6613\)