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Properties

Label 178752gm
Number of curves $6$
Conductor $178752$
CM no
Rank $2$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gm1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 178752gm have rank \(2\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(7\)\(1\)
\(19\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 - T + 11 T^{2}\) 1.11.ab
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - T + 17 T^{2}\) 1.17.ab
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
\(29\) \( 1 + 8 T + 29 T^{2}\) 1.29.i
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 178752gm do not have complex multiplication.

Modular form 178752.2.a.gm

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + q^{9} - 6 q^{11} - 4 q^{13} + q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 178752gm

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
178752.hn6 178752gm1 \([0, 1, 0, -24462433, -46577002753]\) \(52492168638015625/293197968\) \(9042512507628945408\) \([2]\) \(10616832\) \(2.8297\) \(\Gamma_0(N)\)-optimal
178752.hn5 178752gm2 \([0, 1, 0, -24901473, -44818823169]\) \(55369510069623625/3916046302812\) \(120774703573769647030272\) \([2]\) \(21233664\) \(3.1763\)  
178752.hn4 178752gm3 \([0, 1, 0, -35046433, -2466362305]\) \(154357248921765625/89242711068672\) \(2752332618666656957202432\) \([2]\) \(31850496\) \(3.3790\)  
178752.hn3 178752gm4 \([0, 1, 0, -379253793, 2833182710847]\) \(195607431345044517625/752875610010048\) \(23219421221243358321573888\) \([2]\) \(63700992\) \(3.7256\)  
178752.hn2 178752gm5 \([0, 1, 0, -1918198753, 32335314318719]\) \(25309080274342544331625/191933498523648\) \(5919417084886034298175488\) \([2]\) \(95551488\) \(3.9283\)  
178752.hn1 178752gm6 \([0, 1, 0, -30691124193, 2069498717566335]\) \(103665426767620308239307625/5961940992\) \(183872099652156260352\) \([2]\) \(191102976\) \(4.2749\)