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Properties

Label 45570d
Number of curves $2$
Conductor $45570$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 45570d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
45570.e2 45570d1 \([1, 1, 0, -668728, -211067072]\) \(-281115640967896441/468084326400\) \(-55069652916633600\) \([2]\) \(599040\) \(2.1081\) \(\Gamma_0(N)\)-optimal
45570.e1 45570d2 \([1, 1, 0, -10703928, -13483622592]\) \(1152829477932246539641/3188367360\) \(375108231536640\) \([2]\) \(1198080\) \(2.4547\)  

Rank

sage: E.rank()
 

The elliptic curves in class 45570d have rank \(0\).

Complex multiplication

The elliptic curves in class 45570d do not have complex multiplication.

Modular form 45570.2.a.d

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.