Properties

Label 12495b
Number of curves $6$
Conductor $12495$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 12495b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12495.a4 12495b1 \([1, 1, 1, -191346, -32296146]\) \(6585576176607121/187425\) \(22050363825\) \([2]\) \(49152\) \(1.4944\) \(\Gamma_0(N)\)-optimal
12495.a3 12495b2 \([1, 1, 1, -191591, -32209612]\) \(6610905152742241/35128130625\) \(4132789439900625\) \([2, 2]\) \(98304\) \(1.8410\)  
12495.a2 12495b3 \([1, 1, 1, -299636, 8069564]\) \(25288177725059761/14387797265625\) \(1692709960503515625\) \([2, 2]\) \(196608\) \(2.1876\)  
12495.a5 12495b4 \([1, 1, 1, -87466, -66945712]\) \(-629004249876241/16074715228425\) \(-1891174171908972825\) \([2]\) \(196608\) \(2.1876\)  
12495.a1 12495b5 \([1, 1, 1, -3515261, 2530405814]\) \(40832710302042509761/91556816413125\) \(10771567894187743125\) \([2]\) \(393216\) \(2.5342\)  
12495.a6 12495b6 \([1, 1, 1, 1187269, 65761478]\) \(1573196002879828319/926055908203125\) \(-108949551544189453125\) \([2]\) \(393216\) \(2.5342\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12495b have rank \(0\).

Complex multiplication

The elliptic curves in class 12495b do not have complex multiplication.

Modular form 12495.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} + 3 q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} + 2 q^{13} + q^{15} - q^{16} - q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.