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Properties

Label 128778m
Number of curves $2$
Conductor $128778$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 128778m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
128778.k2 128778m1 \([1, 0, 1, -17228878, 27264185600]\) \(117174888570509216929/1273887851544576\) \(6148813346826023337984\) \([]\) \(8692992\) \(2.9960\) \(\Gamma_0(N)\)-optimal
128778.k1 128778m2 \([1, 0, 1, -3779026918, -89416900063840]\) \(1236526859255318155975783969/38367061931916216\) \(185190479836530578634744\) \([]\) \(60850944\) \(3.9689\)  

Rank

sage: E.rank()
 

The elliptic curves in class 128778m have rank \(1\).

Complex multiplication

The elliptic curves in class 128778m do not have complex multiplication.

Modular form 128778.2.a.m

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.