Properties

Label 68970bi
Number of curves $4$
Conductor $68970$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bi1")
 
E.isogeny_class()
 

Elliptic curves in class 68970bi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68970.bd4 68970bi1 \([1, 0, 1, -1213, -27832]\) \(-111284641/123120\) \(-218114590320\) \([2]\) \(122880\) \(0.87032\) \(\Gamma_0(N)\)-optimal
68970.bd3 68970bi2 \([1, 0, 1, -22993, -1343344]\) \(758800078561/324900\) \(575580168900\) \([2, 2]\) \(245760\) \(1.2169\)  
68970.bd2 68970bi3 \([1, 0, 1, -26623, -891772]\) \(1177918188481/488703750\) \(865768504053750\) \([2]\) \(491520\) \(1.5635\)  
68970.bd1 68970bi4 \([1, 0, 1, -367843, -85900564]\) \(3107086841064961/570\) \(1009789770\) \([2]\) \(491520\) \(1.5635\)  

Rank

sage: E.rank()
 

The elliptic curves in class 68970bi have rank \(1\).

Complex multiplication

The elliptic curves in class 68970bi do not have complex multiplication.

Modular form 68970.2.a.bi

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

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.