Elliptic curve isogeny class with Cremona label 198198e (LMFDB label 198198.dg)

• Label: 198198e
• Number of curves: $4$
• Conductor: $198198$
• CM: no
• Rank: $2$

Rank

The elliptic curves in class 198198e have rank \(2\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 + T\)
\(3\)	\(1\)
\(7\)	\(1 + T\)
\(11\)	\(1\)
\(13\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 3 T + 5 T^{2}\)	1.5.d
\(17\)	\( 1 + 4 T + 17 T^{2}\)	1.17.e
\(19\)	\( 1 - 3 T + 19 T^{2}\)	1.19.ad
\(23\)	\( 1 + 5 T + 23 T^{2}\)	1.23.f
\(29\)	\( 1 + 10 T + 29 T^{2}\)	1.29.k
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 198198e do not have complex multiplication.

Modular form 198198.2.a.e

Isogeny 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.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 198198e

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
198198.dg4	198198e1	\([1, -1, 1, 943414, 239170825]\)	\(71903073502287/60782804992\)	\(-78499045713141301248\)	\([2]\)	\(7372800\)	\(2.5051\)	\(\Gamma_0(N)\)-optimal
198198.dg3	198198e2	\([1, -1, 1, -4632266, 2108138761]\)	\(8511781274893233/3440817243136\)	\(4443705256693029110784\)	\([2, 2]\)	\(14745600\)	\(2.8517\)
198198.dg1	198198e3	\([1, -1, 1, -64396586, 198852280201]\)	\(22868021811807457713/8953460393696\)	\(11563107310168513523424\)	\([2]\)	\(29491200\)	\(3.1983\)
198198.dg2	198198e4	\([1, -1, 1, -34078826, -75088962935]\)	\(3389174547561866673/74853681183008\)	\(96671131609592859070752\)	\([2]\)	\(29491200\)	\(3.1983\)