LMFDB - Elliptic curve with Cremona label 324870dd2 (LMFDB label 324870.dd6)

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

$y^2+xy+y=x^3+x^2-426108411x-3375030270711$ y^2+xy+y=x^3+x^2-426108411x-3375030270711	(homogenize, simplify)
$y^2z+xyz+yz^2=x^3+x^2z-426108411xz^2-3375030270711z^3$ y^2z+xyz+yz^2=x^3+x^2z-426108411xz^2-3375030270711z^3	(dehomogenize, simplify)
$y^2=x^3-552236500683x-157457128762773882$ y^2=x^3-552236500683x-157457128762773882	(homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 1, 1, -426108411, -3375030270711])

gp: E = ellinit([1, 1, 1, -426108411, -3375030270711])

magma: E := EllipticCurve([1, 1, 1, -426108411, -3375030270711]);

oscar: E = elliptic_curve([1, 1, 1, -426108411, -3375030270711])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Mordell-Weil group structure

$\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z$

magma: MordellWeilGroup(E);

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(648789, 521991230)$	$7.0107275848039981046344300350$	$\infty$
$(-12461, 6230)$	$0$	$2$
$(23827, -11914)$	$0$	$2$

$\left(-12461, 6230\right)$ , $\left(23827, -11914\right)$ , $\left(648789, 521991230\right)$ , $\left(648789, -522640020\right)$ (-12461, 6230), (23827, -11914), (648789, 521991230), (648789, -522640020)

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

Invariants

Conductor:	$N$	=	$324870$	=	$2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \cdot 17$	comment: Conductor sage: E.conductor().factor() gp: ellglobalred(E)[1] magma: Conductor(E); oscar: conductor(E)
Discriminant:	$\Delta$	=	$31219476031455156900000000$	=	$2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} \cdot 13^{4} \cdot 17^{2}$	comment: Discriminant sage: E.discriminant().factor() gp: E.disc magma: Discriminant(E); oscar: discriminant(E)
j-invariant:	$j$	=	$\frac{72727020009972527154752161}{265361167808100000000}$	=	$2^{-8} \cdot 3^{-8} \cdot 5^{-8} \cdot 7^{-2} \cdot 13^{-4} \cdot 17^{-2} \cdot 73^{3} \cdot 5717977^{3}$	comment: j-invariant sage: E.j_invariant().factor() gp: E.j magma: jInvariant(E); oscar: j_invariant(E)
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$ (no potential complex multiplication)			sage: E.has_cm() magma: HasComplexMultiplication(E);
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$3.7525931670806794651550826271$			gp: ellheight(E) magma: FaltingsHeight(E); oscar: faltings_height(E)
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$2.7796380925530228126024062554$			magma: StableFaltingsHeight(E); oscar: stable_faltings_height(E)
$abc$ quality:	$Q$	≈	$0.9864606854599206$
Szpiro ratio:	$\sigma_{m}$	≈	$5.612103363623219$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$1$	sage: E.analytic_rank() gp: ellanalyticrank(E) magma: AnalyticRank(E);
Mordell-Weil rank:	$r$	=	$1$	comment: Rank sage: E.rank() gp: [lower,upper] = ellrank(E) magma: Rank(E);
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$7.0107275848039981046344300350$	comment: Regulator sage: E.regulator() G = E.gen \\ if available matdet(ellheightmatrix(E,G)) magma: Regulator(E);
Real period:	$\Omega$	≈	$0.033236464944624629787775118256$	comment: Real Period sage: E.period_lattice().omega() gp: if(E.disc>0,2,1)E.omega[1] magma: (Discriminant(E) gt 0 select 2 else 1) RealPeriod(E);
Tamagawa product:	$\prod_{p}c_p$	=	$512$ = $2^{3}\cdot2\cdot2\cdot2^{2}\cdot2\cdot2$	comment: Tamagawa numbers sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] \| i<-[1..#gr[4][,1]]] magma: TamagawaNumbers(E); oscar: tamagawa_numbers(E)
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$4$	comment: Torsion order sage: E.torsion_order() gp: elltors(E)[1] magma: Order(TorsionSubgroup(E)); oscar: prod(torsion_structure(E)[1])
Special value:	$L'(E,1)$	≈	$7.4563776514768313407734389283$	comment: Special L-value r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial() gp: [r,L1r] = ellanalyticrank(E); L1r/r! magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
Analytic order of Ш:	Ш ${}_{\mathrm{an}}$	≈	$1$ (rounded)	comment: Order of Sha sage: E.sha().an_numerical() magma: MordellWeilShaInformation(E);

BSD formula

$\displaystyle 7.456377651 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.033236 \cdot 7.010728 \cdot 512}{4^2} \approx 7.456377651$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

Modular invariants

Modular form 324870.2.a.dd

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

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

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

Modular degree:	150994944	comment: Modular degree sage: E.modular_degree() gp: ellmoddegree(E) magma: ModularDegree(E);
$\Gamma_0(N)$ -optimal:	no
Manin constant:	1	comment: Manin constant magma: ManinConstant(E);

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 6 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$8$	$I_{8}$	split multiplicative	-1	1	8	8
$3$	$2$	$I_{8}$	nonsplit multiplicative	1	1	8	8
$5$	$2$	$I_{8}$	nonsplit multiplicative	1	1	8	8
$7$	$4$	$I_{2}^{*}$	additive	-1	2	8	2
$13$	$2$	$I_{4}$	nonsplit multiplicative	1	1	4	4
$17$	$2$	$I_{2}$	nonsplit multiplicative	1	1	2	2

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

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
$2$	2Cs	8.48.0.98

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

gens = [[159111, 371264, 211966, 370935], [297033, 8, 74212, 371241], [1, 16, 0, 1], [371265, 16, 371264, 17], [5, 8, 68, 92929], [114249, 16, 285668, 121], [1, 16, 4, 65], [1, 0, 16, 1], [123761, 8, 0, 1], [15, 16, 224, 232289], [65521, 16, 21846, 97]]

GL(2,Integers(371280)).subgroup(gens)

Gens := [[159111, 371264, 211966, 370935], [297033, 8, 74212, 371241], [1, 16, 0, 1], [371265, 16, 371264, 17], [5, 8, 68, 92929], [114249, 16, 285668, 121], [1, 16, 4, 65], [1, 0, 16, 1], [123761, 8, 0, 1], [15, 16, 224, 232289], [65521, 16, 21846, 97]];

sub<GL(2,Integers(371280))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $371280 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17$ , index $768$ , genus $13$ , and generators

$\left(\begin{array}{rr} 159111 & 371264 \\ 211966 & 370935 \end{array}\right),\left(\begin{array}{rr} 297033 & 8 \\ 74212 & 371241 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 371265 & 16 \\ 371264 & 17 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 68 & 92929 \end{array}\right),\left(\begin{array}{rr} 114249 & 16 \\ 285668 & 121 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 123761 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 224 & 232289 \end{array}\right),\left(\begin{array}{rr} 65521 & 16 \\ 21846 & 97 \end{array}\right)$ .

The torsion field $K:=\Q(E[371280])$ is a degree- $3051534277662474240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/371280\Z)$ .

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	$4$	$49 = 7^{2}$
$3$	nonsplit multiplicative	$4$	$108290 = 2 \cdot 5 \cdot 7^{2} \cdot 13 \cdot 17$
$5$	nonsplit multiplicative	$6$	$64974 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \cdot 17$
$7$	additive	$32$	$6630 = 2 \cdot 3 \cdot 5 \cdot 13 \cdot 17$
$13$	nonsplit multiplicative	$14$	$24990 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17$
$17$	nonsplit multiplicative	$18$	$19110 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13$

Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 324870dd consists of 8 curves linked by isogenies of degrees dividing 16.

Twists

The minimal quadratic twist of this elliptic curve is 46410cn2, its twist by $-7$ .

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	$\Q(\sqrt{-7})$	$\Z/2\Z \oplus \Z/8\Z$	not in database
$4$	$\Q(\sqrt{7}, \sqrt{-17})$	$\Z/2\Z \oplus \Z/4\Z$	not in database
$4$	$\Q(\sqrt{7}, \sqrt{17})$	$\Z/2\Z \oplus \Z/4\Z$	not in database
$8$	8.0.51336683776.1	$\Z/4\Z \oplus \Z/8\Z$	not in database
$8$	deg 8	$\Z/2\Z \oplus \Z/16\Z$	not in database
$8$	deg 8	$\Z/2\Z \oplus \Z/16\Z$	not in database
$8$	deg 8	$\Z/2\Z \oplus \Z/6\Z$	not in database
$16$	deg 16	$\Z/2\Z \oplus \Z/8\Z$	not in database
$16$	deg 16	$\Z/2\Z \oplus \Z/8\Z$	not in database
$16$	deg 16	$\Z/2\Z \oplus \Z/24\Z$	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

No Iwasawa invariant data is available for this curve.

$p$ -adic regulators

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

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Database

Properties

Related objects

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

Mordell-Weil group structure

Mordell-Weil generators

Integral points

Invariants

BSD invariants

BSD formula

Modular invariants

Local data at primes of bad reduction

Galois representations

Isogenies

Twists

Growth of torsion in number fields

Iwasawa invariants

$p$ -adic regulators

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	324870dd2

Conductor	$324870$
Discriminant	$3.122\times 10^{25}$
j-invariant	$\frac{72727020009972527154752161}{265361167808100000000}$
CM	no
Rank	$1$
Torsion structure	$\Z/{2}\Z \oplus \Z/{2}\Z$

Properties

Related objects

Downloads

Learn more

Mordell-Weil group structure

Invariants

Local data at primes of bad reduction

Growth of torsion in number fields

ppp-adic regulators

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

$p$ -adic regulators