Elliptic curve with LMFDB label 42978.r2 (Cremona label 42978v2)

• Label: 42978.r2
• Conductor: $42978$
• Discriminant: $-8.828\times 10^{22}$
• j-invariant: $\frac{6612600178387895114470319}{88280159906925363790998}$
• CM: no
• Rank: $2$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2+xy=x^3+3910355x-13981592329$

(, )

Mordell-Weil group structure

$\Z \oplus \Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(377987/196, 71637061/2744)$	$5.4407761040444123211678257951$	$\infty$
$(1327261/400, 1494671681/8000)$	$7.1684564494663214315533488117$	$\infty$

Integral points

None

Invariants

Conductor:	$N$	=	$42978$	=	$2 \cdot 3 \cdot 13 \cdot 19 \cdot 29$
Discriminant:	$\Delta$	=	$-88280159906925363790998$	=	$-1 \cdot 2 \cdot 3^{3} \cdot 13 \cdot 19^{10} \cdot 29^{5}$
j-invariant:	$j$	=	$\frac{6612600178387895114470319}{88280159906925363790998}$	=	$2^{-1} \cdot 3^{-3} \cdot 13^{-1} \cdot 19^{-10} \cdot 29^{-5} \cdot 271^{3} \cdot 659^{3} \cdot 1051^{3}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$    (no potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$3.0836697317025411774210830567$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$3.0836697317025411774210830567$
$abc$ quality:	$Q$	≈	$1.0074887199845604$
Szpiro ratio:	$\sigma_{m}$	≈	$5.647047417800233$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$2$
Mordell-Weil rank:	$r$	=	$2$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$37.965941055531387170118891952$
Real period:	$\Omega$	≈	$0.052695131410377958484561407502$
Tamagawa product:	$\prod_{p}c_p$	=	$6$  = $1\cdot3\cdot1\cdot2\cdot1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(2)}(E,1)/2!$	≈	$12.003721518239340623131796423$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$\begin{aligned} 12.003721518 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.052695 \cdot 37.965941 \cdot 6}{1^2} \\ & \approx 12.003721518\end{aligned}$

Modular invariants

Modular form   42978.2.a.r

$q + q^{2} + q^{3} + q^{4} - 4 q^{5} + q^{6} - 2 q^{7} + q^{8} + q^{9} - 4 q^{10} - 3 q^{11} + q^{12} + q^{13} - 2 q^{14} - 4 q^{15} + q^{16} - 7 q^{17} + q^{18} - q^{19} + O(q^{20})$

For more coefficients, see the Downloads section to the right.

Modular degree:	9000000
$\Gamma_0(N)$-optimal:	no
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is semistable. There are 5 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$1$	$I_{1}$	split multiplicative	-1	1	1	1
$3$	$3$	$I_{3}$	split multiplicative	-1	1	3	3
$13$	$1$	$I_{1}$	split multiplicative	-1	1	1	1
$19$	$2$	$I_{10}$	nonsplit multiplicative	1	1	10	10
$29$	$1$	$I_{5}$	nonsplit multiplicative	1	1	5	5

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$	mod-$\ell$ image	$\ell$-adic image
$5$	5B.1.2	5.24.0.3

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $45240 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 29$, index $48$, genus $1$, and generators

$\left(\begin{array}{rr} 6 & 5 \\ 30155 & 45236 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 45185 & 45121 \end{array}\right),\left(\begin{array}{rr} 45231 & 10 \\ 45230 & 11 \end{array}\right),\left(\begin{array}{rr} 17406 & 5 \\ 38275 & 45236 \end{array}\right),\left(\begin{array}{rr} 33937 & 22630 \\ 45200 & 42921 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11311 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6 & 5 \\ 22615 & 45236 \end{array}\right),\left(\begin{array}{rr} 43686 & 5 \\ 38995 & 45236 \end{array}\right)$.

The torsion field $K:=\Q(E[45240])$ is a degree-$13179582362419200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/45240\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$	Reduction type	Serre weight	Serre conductor
$2$	split multiplicative	$4$	$1131 = 3 \cdot 13 \cdot 29$
$3$	split multiplicative	$4$	$14326 = 2 \cdot 13 \cdot 19 \cdot 29$
$5$	good	$2$	$78 = 2 \cdot 3 \cdot 13$
$13$	split multiplicative	$14$	$3306 = 2 \cdot 3 \cdot 19 \cdot 29$
$19$	nonsplit multiplicative	$20$	$2262 = 2 \cdot 3 \cdot 13 \cdot 29$
$29$	nonsplit multiplicative	$30$	$1482 = 2 \cdot 3 \cdot 13 \cdot 19$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 5.
Its isogeny class 42978.r consists of 2 curves linked by isogenies of degree 5.

Twists

This elliptic curve is its own minimal quadratic twist.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.9048.1	$\Z/2\Z$	not in database
$4$	$\Q(\zeta_{5})$	$\Z/5\Z$	not in database
$5$	5.1.115672050000.7	$\Z/5\Z$	not in database
$6$	6.0.740726318592.1	$\Z/2\Z \oplus \Z/2\Z$	not in database
$8$	deg 8	$\Z/3\Z$	not in database
$12$	deg 12	$\Z/4\Z$	not in database
$12$	deg 12	$\Z/10\Z$	not in database
$15$	15.1.2535536379673377465657033505536000000000000000.1	$\Z/10\Z$	not in database
$20$	20.0.22378127440839359772149500781250000000000000000.11	$\Z/5\Z \oplus \Z/5\Z$	not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	split	split	ord	ord	ord	split	ord	nonsplit	ord	nonsplit	ord	ord	ord	ord	ord
$\lambda$-invariant(s)	12	3	4	2	2	3	2	2	2	2	2	2	2	2	2
$\mu$-invariant(s)	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0

$p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.