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Properties

Label 3150l
Number of curves $6$
Conductor $3150$
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, -1, 0, -117, -959]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -117, -959]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -117, -959]) 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 3150l have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1\)
\(7\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 3150l do not have complex multiplication.

Modular form 3150.2.a.l

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);
 
\(q - q^{2} + q^{4} - q^{7} - q^{8} + 4 q^{13} + q^{14} + q^{16} + 6 q^{17} + 2 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 3150l

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
3150.i5 3150l1 \([1, -1, 0, -117, -959]\) \(-15625/28\) \(-318937500\) \([2]\) \(1152\) \(0.32194\) \(\Gamma_0(N)\)-optimal
3150.i4 3150l2 \([1, -1, 0, -2367, -43709]\) \(128787625/98\) \(1116281250\) \([2]\) \(2304\) \(0.66851\)  
3150.i6 3150l3 \([1, -1, 0, 1008, 20416]\) \(9938375/21952\) \(-250047000000\) \([2]\) \(3456\) \(0.87125\)  
3150.i3 3150l4 \([1, -1, 0, -7992, 227416]\) \(4956477625/941192\) \(10720765125000\) \([2]\) \(6912\) \(1.2178\)  
3150.i2 3150l5 \([1, -1, 0, -38367, 2910541]\) \(-548347731625/1835008\) \(-20901888000000\) \([2]\) \(10368\) \(1.4206\)  
3150.i1 3150l6 \([1, -1, 0, -614367, 185502541]\) \(2251439055699625/25088\) \(285768000000\) \([2]\) \(20736\) \(1.7671\)