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Properties

Label 7350w
Number of curves $8$
Conductor $7350$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("w1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 7350.2.a.w

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

Isogeny matrix

Copy content 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 7350w

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7350.bd7 7350w1 \([1, 0, 1, -50251, 1521398]\) \(7633736209/3870720\) \(7115411520000000\) \([2]\) \(55296\) \(1.7342\) \(\Gamma_0(N)\)-optimal
7350.bd5 7350w2 \([1, 0, 1, -442251, -112158602]\) \(5203798902289/57153600\) \(105063498225000000\) \([2, 2]\) \(110592\) \(2.0808\)  
7350.bd4 7350w3 \([1, 0, 1, -3284251, 2290605398]\) \(2131200347946769/2058000\) \(3783150656250000\) \([2]\) \(165888\) \(2.2836\)  
7350.bd2 7350w4 \([1, 0, 1, -7057251, -7216668602]\) \(21145699168383889/2593080\) \(4766769826875000\) \([2]\) \(221184\) \(2.4274\)  
7350.bd6 7350w5 \([1, 0, 1, -99251, -281600602]\) \(-58818484369/18600435000\) \(-34192540270546875000\) \([2]\) \(221184\) \(2.4274\)  
7350.bd3 7350w6 \([1, 0, 1, -3308751, 2254688398]\) \(2179252305146449/66177562500\) \(121651938290039062500\) \([2, 2]\) \(331776\) \(2.6301\)  
7350.bd1 7350w7 \([1, 0, 1, -7902501, -5380124102]\) \(29689921233686449/10380965400750\) \(19082971850513074218750\) \([2]\) \(663552\) \(2.9767\)  
7350.bd8 7350w8 \([1, 0, 1, 892999, 7590910898]\) \(42841933504271/13565917968750\) \(-24937760673522949218750\) \([2]\) \(663552\) \(2.9767\)