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Properties

Label 34307b
Number of curves $2$
Conductor $34307$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 34307b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
34307.c2 34307b1 \([1, 1, 1, -1609, 25462]\) \(-95443993/5887\) \(-28415424583\) \([2]\) \(27648\) \(0.76027\) \(\Gamma_0(N)\)-optimal
34307.c1 34307b2 \([1, 1, 1, -26114, 1613386]\) \(408023180713/1421\) \(6858895589\) \([2]\) \(55296\) \(1.1068\)  

Rank

sage: E.rank()
 

The elliptic curves in class 34307b have rank \(1\).

Complex multiplication

The elliptic curves in class 34307b do not have complex multiplication.

Modular form 34307.2.a.b

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