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Properties

Label 7350bc
Number of curves $8$
Conductor $7350$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("bc1")
 
E.isogeny_class()
 

Elliptic curves in class 7350bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7350.w7 7350bc1 \([1, 0, 1, 257224, -37815802]\) \(1023887723039/928972800\) \(-1707698764800000000\) \([2]\) \(147456\) \(2.1858\) \(\Gamma_0(N)\)-optimal
7350.w6 7350bc2 \([1, 0, 1, -1310776, -338871802]\) \(135487869158881/51438240000\) \(94557148402500000000\) \([2, 2]\) \(294912\) \(2.5324\)  
7350.w4 7350bc3 \([1, 0, 1, -18460776, -30522871802]\) \(378499465220294881/120530818800\) \(221567660953143750000\) \([2]\) \(589824\) \(2.8790\)  
7350.w5 7350bc4 \([1, 0, 1, -9248776, 10583816198]\) \(47595748626367201/1215506250000\) \(2234423356347656250000\) \([2, 2]\) \(589824\) \(2.8790\)  
7350.w2 7350bc5 \([1, 0, 1, -147061276, 686416316198]\) \(191342053882402567201/129708022500\) \(238437799048476562500\) \([2, 2]\) \(1179648\) \(3.2255\)  
7350.w8 7350bc6 \([1, 0, 1, 1555724, 33835100198]\) \(226523624554079/269165039062500\) \(-494796838760375976562500\) \([2]\) \(1179648\) \(3.2255\)  
7350.w1 7350bc7 \([1, 0, 1, -2352980026, 43931247491198]\) \(783736670177727068275201/360150\) \(662051364843750\) \([2]\) \(2359296\) \(3.5721\)  
7350.w3 7350bc8 \([1, 0, 1, -146142526, 695416391198]\) \(-187778242790732059201/4984939585440150\) \(-9163643082616378239843750\) \([2]\) \(2359296\) \(3.5721\)  

Rank

sage: E.rank()
 

The elliptic curves in class 7350bc have rank \(1\).

Complex multiplication

The elliptic curves in class 7350bc do not have complex multiplication.

Modular form 7350.2.a.bc

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} - 4 q^{11} + q^{12} - 2 q^{13} + q^{16} + 2 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.