Properties

Label 5070.a
Number of curves $6$
Conductor $5070$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 5070.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5070.a1 5070a5 \([1, 1, 0, -1523538, -724450878]\) \(81025909800741361/11088090\) \(53520092604810\) \([2]\) \(86016\) \(2.0465\)  
5070.a2 5070a4 \([1, 1, 0, -142808, 20696148]\) \(66730743078481/60937500\) \(294133673437500\) \([2]\) \(43008\) \(1.7000\)  
5070.a3 5070a3 \([1, 1, 0, -95488, -11282708]\) \(19948814692561/231344100\) \(1116653783976900\) \([2, 2]\) \(43008\) \(1.7000\)  
5070.a4 5070a6 \([1, 1, 0, -19438, -28667738]\) \(-168288035761/73415764890\) \(-354363874712936010\) \([2]\) \(86016\) \(2.0465\)  
5070.a5 5070a2 \([1, 1, 0, -10988, 158592]\) \(30400540561/15210000\) \(73415764890000\) \([2, 2]\) \(21504\) \(1.3534\)  
5070.a6 5070a1 \([1, 1, 0, 2532, 20688]\) \(371694959/249600\) \(-1204771526400\) \([2]\) \(10752\) \(1.0068\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 5070.a have rank \(1\).

Complex multiplication

The elliptic curves in class 5070.a do not have complex multiplication.

Modular form 5070.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} - 4q^{11} - q^{12} + q^{15} + q^{16} - 6q^{17} - q^{18} - 4q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.