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Properties

Label 7605q
Number of curves $8$
Conductor $7605$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("q1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 7605q have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(5\)\(1 - T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 + 4 T + 17 T^{2}\) 1.17.e
\(19\) \( 1 + 6 T + 19 T^{2}\) 1.19.g
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 4 T + 29 T^{2}\) 1.29.e
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 7605q do not have complex multiplication.

Modular form 7605.2.a.q

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + q^{5} + 3 q^{8} - q^{10} - 4 q^{11} - q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 7605q

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7605.g7 7605q1 \([1, -1, 1, -32, -11046]\) \(-1/15\) \(-52781156415\) \([2]\) \(4608\) \(0.73636\) \(\Gamma_0(N)\)-optimal
7605.g6 7605q2 \([1, -1, 1, -7637, -251364]\) \(13997521/225\) \(791717346225\) \([2, 2]\) \(9216\) \(1.0829\)  
7605.g4 7605q3 \([1, -1, 1, -121712, -16313124]\) \(56667352321/15\) \(52781156415\) \([2]\) \(18432\) \(1.4295\)  
7605.g5 7605q4 \([1, -1, 1, -15242, 338784]\) \(111284641/50625\) \(178136402900625\) \([2, 2]\) \(18432\) \(1.4295\)  
7605.g2 7605q5 \([1, -1, 1, -205367, 35854134]\) \(272223782641/164025\) \(577161945398025\) \([2, 2]\) \(36864\) \(1.7761\)  
7605.g8 7605q6 \([1, -1, 1, 53203, 2501646]\) \(4733169839/3515625\) \(-12370583534765625\) \([2]\) \(36864\) \(1.7761\)  
7605.g1 7605q7 \([1, -1, 1, -3285392, 2292896454]\) \(1114544804970241/405\) \(1425091223205\) \([2]\) \(73728\) \(2.1227\)  
7605.g3 7605q8 \([1, -1, 1, -167342, 49512714]\) \(-147281603041/215233605\) \(-757351904751288405\) \([2]\) \(73728\) \(2.1227\)