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Properties

Label 3570.t
Number of curves $6$
Conductor $3570$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("t1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 3570.t have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 3570.t do not have complex multiplication.

Modular form 3570.2.a.t

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - q^{7} + q^{8} + q^{9} + q^{10} + 4 q^{11} - q^{12} + 6 q^{13} - q^{14} - q^{15} + q^{16} + q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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 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 sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 3570.t

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3570.t1 3570t5 \([1, 1, 1, -1742580, -886120275]\) \(585196747116290735872321/836876053125000\) \(836876053125000\) \([2]\) \(73728\) \(2.1347\)  
3570.t2 3570t4 \([1, 1, 1, -252620, 48757805]\) \(1782900110862842086081/328139630024640\) \(328139630024640\) \([4]\) \(36864\) \(1.7881\)  
3570.t3 3570t3 \([1, 1, 1, -109900, -13616083]\) \(146796951366228945601/5397929064360000\) \(5397929064360000\) \([2, 2]\) \(36864\) \(1.7881\)  
3570.t4 3570t2 \([1, 1, 1, -17420, 588845]\) \(584614687782041281/184812061593600\) \(184812061593600\) \([2, 4]\) \(18432\) \(1.4415\)  
3570.t5 3570t1 \([1, 1, 1, 3060, 64557]\) \(3168685387909439/3563732336640\) \(-3563732336640\) \([4]\) \(9216\) \(1.0949\) \(\Gamma_0(N)\)-optimal
3570.t6 3570t6 \([1, 1, 1, 43100, -48377683]\) \(8854313460877886399/1016927675429790600\) \(-1016927675429790600\) \([2]\) \(73728\) \(2.1347\)