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Properties

Label 3120r
Number of curves $8$
Conductor $3120$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("r1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 3120r have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 3120r do not have complex multiplication.

Modular form 3120.2.a.r

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{9} - 4 q^{11} + q^{13} - q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 3120r

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3120.k6 3120r1 \([0, -1, 0, -1760, -27840]\) \(147281603041/5265\) \(21565440\) \([2]\) \(1536\) \(0.49627\) \(\Gamma_0(N)\)-optimal
3120.k5 3120r2 \([0, -1, 0, -1840, -25088]\) \(168288035761/27720225\) \(113542041600\) \([2, 2]\) \(3072\) \(0.84284\)  
3120.k4 3120r3 \([0, -1, 0, -8320, 270400]\) \(15551989015681/1445900625\) \(5922408960000\) \([2, 4]\) \(6144\) \(1.1894\)  
3120.k7 3120r4 \([0, -1, 0, 3360, -145728]\) \(1023887723039/2798036865\) \(-11460758999040\) \([2]\) \(6144\) \(1.1894\)  
3120.k2 3120r5 \([0, -1, 0, -130000, 18084352]\) \(59319456301170001/594140625\) \(2433600000000\) \([2, 4]\) \(12288\) \(1.5360\)  
3120.k8 3120r6 \([0, -1, 0, 9680, 1264000]\) \(24487529386319/183539412225\) \(-751777432473600\) \([4]\) \(12288\) \(1.5360\)  
3120.k1 3120r7 \([0, -1, 0, -2080000, 1155324352]\) \(242970740812818720001/24375\) \(99840000\) \([4]\) \(24576\) \(1.8826\)  
3120.k3 3120r8 \([0, -1, 0, -126880, 18990400]\) \(-55150149867714721/5950927734375\) \(-24375000000000000\) \([4]\) \(24576\) \(1.8826\)