Properties

Label 100048.b
Number of curves $3$
Conductor $100048$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 100048.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
100048.b1 100048i3 \([0, -1, 0, -5065493, 4389831133]\) \(727057727488000/37\) \(731512557568\) \([]\) \(933120\) \(2.1977\)  
100048.b2 100048i2 \([0, -1, 0, -63093, 5927869]\) \(1404928000/50653\) \(1001440691310592\) \([]\) \(311040\) \(1.6484\)  
100048.b3 100048i1 \([0, -1, 0, -9013, -323779]\) \(4096000/37\) \(731512557568\) \([]\) \(103680\) \(1.0991\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 100048.b have rank \(0\).

Complex multiplication

The elliptic curves in class 100048.b do not have complex multiplication.

Modular form 100048.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{7} - 2 q^{9} + 3 q^{11} + 6 q^{17} + 2 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.