Properties

Label 350.f
Number of curves $6$
Conductor $350$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 350.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
350.f1 350d6 \([1, 1, 1, -68263, -6893219]\) \(2251439055699625/25088\) \(392000000\) \([2]\) \(864\) \(1.2178\)  
350.f2 350d5 \([1, 1, 1, -4263, -109219]\) \(-548347731625/1835008\) \(-28672000000\) \([2]\) \(432\) \(0.87125\)  
350.f3 350d4 \([1, 1, 1, -888, -8719]\) \(4956477625/941192\) \(14706125000\) \([2]\) \(288\) \(0.66851\)  
350.f4 350d2 \([1, 1, 1, -263, 1531]\) \(128787625/98\) \(1531250\) \([2]\) \(96\) \(0.11921\)  
350.f5 350d1 \([1, 1, 1, -13, 31]\) \(-15625/28\) \(-437500\) \([2]\) \(48\) \(-0.22737\) \(\Gamma_0(N)\)-optimal
350.f6 350d3 \([1, 1, 1, 112, -719]\) \(9938375/21952\) \(-343000000\) \([2]\) \(144\) \(0.32194\)  

Rank

sage: E.rank()
 

The elliptic curves in class 350.f have rank \(0\).

Complex multiplication

The elliptic curves in class 350.f do not have complex multiplication.

Modular form 350.2.a.f

sage: E.q_eigenform(10)
 
\(q + q^{2} + 2q^{3} + q^{4} + 2q^{6} - q^{7} + q^{8} + q^{9} + 2q^{12} + 4q^{13} - q^{14} + q^{16} - 6q^{17} + q^{18} + 2q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.