Properties

Label 42a
Number of curves $6$
Conductor $42$
CM no
Rank $0$
Graph

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

Elliptic curves in class 42a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42.a5 42a1 \([1, 1, 1, -4, 5]\) \(-7189057/16128\) \(-16128\) \([8]\) \(4\) \(-0.50405\) \(\Gamma_0(N)\)-optimal
42.a4 42a2 \([1, 1, 1, -84, 261]\) \(65597103937/63504\) \(63504\) \([2, 4]\) \(8\) \(-0.15747\)  
42.a3 42a3 \([1, 1, 1, -104, 101]\) \(124475734657/63011844\) \(63011844\) \([2, 2]\) \(16\) \(0.18910\)  
42.a1 42a4 \([1, 1, 1, -1344, 18405]\) \(268498407453697/252\) \(252\) \([4]\) \(16\) \(0.18910\)  
42.a2 42a5 \([1, 1, 1, -914, -10915]\) \(84448510979617/933897762\) \(933897762\) \([2]\) \(32\) \(0.53568\)  
42.a6 42a6 \([1, 1, 1, 386, 1277]\) \(6359387729183/4218578658\) \(-4218578658\) \([2]\) \(32\) \(0.53568\)  

Rank

sage: E.rank()
 

The elliptic curves in class 42a have rank \(0\).

Complex multiplication

The elliptic curves in class 42a do not have complex multiplication.

Modular form 42.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - 2q^{10} - 4q^{11} - q^{12} + 6q^{13} - q^{14} + 2q^{15} + q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20})\)  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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.