Elliptic curve isogeny class with Cremona label 60690a (LMFDB label 60690.d)

• Label: 60690a
• Number of curves: $8$
• Conductor: $60690$
• CM: no
• Rank: $1$

Elliptic curves in class 60690a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
60690.d7	60690a1	$[1, 1, 0, -143783, 20917173]$	$13619385906841/6048000$	$145984017312000$	$[2]$	$442368$	$1.6764$	$\Gamma_0(N)$-optimal
60690.d6	60690a2	$[1, 1, 0, -166903, 13708357]$	$21302308926361/8930250000$	$215554525562250000$	$[2, 2]$	$884736$	$2.0230$
60690.d5	60690a3	$[1, 1, 0, -425558, -81384492]$	$353108405631241/86318776320$	$2083525419419566080$	$[2]$	$1327104$	$2.2257$
60690.d8	60690a4	$[1, 1, 0, 555597, 101419857]$	$785793873833639/637994920500$	$-15399646415218264500$	$[2]$	$1769472$	$2.3696$
60690.d4	60690a5	$[1, 1, 0, -1259323, -534904967]$	$9150443179640281/184570312500$	$4455078653320312500$	$[2]$	$1769472$	$2.3696$
60690.d2	60690a6	$[1, 1, 0, -6344278, -6152807468]$	$1169975873419524361/108425318400$	$2617123604226969600$	$[2, 2]$	$2654208$	$2.5723$
60690.d3	60690a7	$[1, 1, 0, -5881878, -7087132908]$	$-932348627918877961/358766164249920$	$-8659743044447777244480$	$[2]$	$5308416$	$2.9189$
60690.d1	60690a8	$[1, 1, 0, -101506198, -393671178092]$	$4791901410190533590281/41160000$	$993502340040000$	$[2]$	$5308416$	$2.9189$

Rank

The elliptic curves in class 60690a have rank $1$.

L-function data

Bad L-factors:

Prime	L-Factor
$2$	$1 + T$
$3$	$1 + T$
$5$	$1 + T$
$7$	$1 + T$
$17$	$1$

Good L-factors:

Prime	L-Factor	Isogeny Class over $\mathbb{F}_p$
$11$	$1 + 4 T + 11 T^{2}$	1.11.e
$13$	$1 - 4 T + 13 T^{2}$	1.13.ae
$19$	$1 + 6 T + 19 T^{2}$	1.19.g
$23$	$1 + 2 T + 23 T^{2}$	1.23.c
$29$	$1 + 29 T^{2}$	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 60690a do not have complex multiplication.

Modular form 60690.2.a.a

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.

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

Isogeny graph

The vertices are labelled with Cremona labels.