Properties

Label 230115b
Number of curves $4$
Conductor $230115$
CM no
Rank $1$
Graph

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

Elliptic curves in class 230115b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
230115.b4 230115b1 \([1, 1, 1, -2777261, 1732668458]\) \(16003198512756001/488525390625\) \(72319290500244140625\) \([2]\) \(6488064\) \(2.5866\) \(\Gamma_0(N)\)-optimal
230115.b2 230115b2 \([1, 1, 1, -44105386, 112723480958]\) \(64096096056024006001/62562515625\) \(9261497618623265625\) \([2, 2]\) \(12976128\) \(2.9332\)  
230115.b1 230115b3 \([1, 1, 1, -705686011, 7215188438708]\) \(262537424941059264096001/250125\) \(37027476736125\) \([2]\) \(25952256\) \(3.2797\)  
230115.b3 230115b4 \([1, 1, 1, -43774761, 114497085708]\) \(-62665433378363916001/2004003001000125\) \(-296664365811721393486125\) \([2]\) \(25952256\) \(3.2797\)  

Rank

sage: E.rank()
 

The elliptic curves in class 230115b have rank \(1\).

Complex multiplication

The elliptic curves in class 230115b do not have complex multiplication.

Modular form 230115.2.a.b

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