Properties

Label 42978.c
Number of curves 44
Conductor 4297842978
CM no
Rank 11
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 42978.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42978.c1 42978b4 [1,1,0,16059278499,783322981022457][1, 1, 0, -16059278499, -783322981022457] 458038307459437803276572539343003833/86000447460798458038307459437803276572539343003833/86000447460798 8600044746079886000447460798 [2][2] 4091904040919040 3.96753.9675  
42978.c2 42978b2 [1,1,0,1003704909,12239735151735][1, 1, 0, -1003704909, -12239735151735] 111825759760338976846738658338393/1532291201797601099556111825759760338976846738658338393/1532291201797601099556 15322912017976010995561532291201797601099556 [2,2][2, 2] 2045952020459520 3.62093.6209  
42978.c3 42978b3 [1,1,0,1002813239,12262566362085][1, 1, 0, -1002813239, -12262566362085] 111527993597885114164012178708473/413980765601504764798430334-111527993597885114164012178708473/413980765601504764798430334 413980765601504764798430334-413980765601504764798430334 [2][2] 4091904040919040 3.96753.9675  
42978.c4 42978b1 [1,1,0,62787289,190908660587][1, 1, 0, -62787289, -190908660587] 27374041292637614212993237273/10105156631438781237748827374041292637614212993237273/101051566314387812377488 101051566314387812377488101051566314387812377488 [2][2] 1022976010229760 3.27433.2743 Γ0(N)\Gamma_0(N)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 42978.c have rank 11.

Complex multiplication

The elliptic curves in class 42978.c do not have complex multiplication.

Modular form 42978.2.a.c

sage: E.q_eigenform(10)
 
qq2q3+q4+2q5+q6+4q7q8+q92q104q11q12+q134q142q15+q16+2q17q18q19+O(q20)q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} + q^{13} - 4 q^{14} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - q^{19} + O(q^{20}) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The i,ji,j entry is the smallest degree of a cyclic isogeny between the ii-th and jj-th curve in the isogeny class, in the LMFDB numbering.

(1244212242144241)\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)

Isogeny graph

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

The vertices are labelled with LMFDB labels.