Properties

Label 8349c
Number of curves $6$
Conductor $8349$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 8349c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8349.d6 8349c1 \([1, 0, 1, 3748, 259301]\) \(3288008303/18259263\) \(-32347398219543\) \([2]\) \(15360\) \(1.2743\) \(\Gamma_0(N)\)-optimal
8349.d5 8349c2 \([1, 0, 1, -45257, 3336815]\) \(5786435182177/627352209\) \(1111392706728249\) \([2, 2]\) \(30720\) \(1.6209\)  
8349.d4 8349c3 \([1, 0, 1, -170492, -23513569]\) \(309368403125137/44372288367\) \(78608215551730887\) \([2]\) \(61440\) \(1.9675\)  
8349.d2 8349c4 \([1, 0, 1, -704102, 227344115]\) \(21790813729717297/304746849\) \(539877632561289\) \([2, 2]\) \(61440\) \(1.9675\)  
8349.d1 8349c5 \([1, 0, 1, -11265587, 14552942369]\) \(89254274298475942657/17457\) \(30926140377\) \([2]\) \(122880\) \(2.3141\)  
8349.d3 8349c6 \([1, 0, 1, -684137, 240848441]\) \(-19989223566735457/2584262514273\) \(-4578178684047990153\) \([2]\) \(122880\) \(2.3141\)  

Rank

sage: E.rank()
 

The elliptic curves in class 8349c have rank \(1\).

Complex multiplication

The elliptic curves in class 8349c do not have complex multiplication.

Modular form 8349.2.a.c

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.