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Properties

Label 444030.a2
Conductor $444030$
Discriminant $-9.051\times 10^{21}$
j-invariant \( -\frac{119305480789133569}{192379221760500} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3+x^2-3702423x+5334137433\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3+x^2z-3702423xz^2+5334137433z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-4798340883x+248941491183918\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 1, 0, -3702423, 5334137433])
 
Copy content gp:E = ellinit([1, 1, 0, -3702423, 5334137433])
 
Copy content magma:E := EllipticCurve([1, 1, 0, -3702423, 5334137433]);
 
Copy content oscar:E = elliptic_curve([1, 1, 0, -3702423, 5334137433])
 
Copy content comment:Simplified equation
 
Copy content sage:E.short_weierstrass_model()
 
Copy content magma:WeierstrassModel(E);
 
Copy content oscar:short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z\)

Copy content comment:Mordell-Weil group
 
Copy content magma:MordellWeilGroup(E);
 

Mordell-Weil generators

$P$$\hat{h}(P)$Order
\( \left(1271, 51168\right) \)$1.7916313045745017576164402296$$\infty$
\( \left(-\frac{9717}{4}, \frac{9717}{8}\right) \)$0$$2$

$P$$\hat{h}(P)$Order
\([1271:51168:1]\)$1.7916313045745017576164402296$$\infty$
\([-19434:9717:8]\)$0$$2$

$P$$\hat{h}(P)$Order
\( \left(45771, 11189556\right) \)$1.7916313045745017576164402296$$\infty$
\( \left(-87438, 0\right) \)$0$$2$

Integral points

\( \left(1271, 51168\right) \), \( \left(1271, -52439\right) \), \( \left(1353, 52275\right) \), \( \left(1353, -53628\right) \), \( \left(1629381, 2079044031\right) \), \( \left(1629381, -2080673412\right) \) Copy content Toggle raw display

\([1271:51168:1]\), \([1271:-52439:1]\), \([1353:52275:1]\), \([1353:-53628:1]\), \([1629381:2079044031:1]\), \([1629381:-2080673412:1]\) Copy content Toggle raw display

\((45771,\pm 11189556)\), \((48723,\pm 11437524)\), \((58657731,\pm 449249483844)\) Copy content Toggle raw display

Copy content comment:Integral points
 
Copy content sage:E.integral_points()
 
Copy content magma:IntegralPoints(E);
 

Invariants

Conductor: $N$  =  \( 444030 \) = $2 \cdot 3 \cdot 5 \cdot 19^{2} \cdot 41$
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Minimal Discriminant: $\Delta$  =  $-9050649973817093500500$ = $-1 \cdot 2^{2} \cdot 3^{4} \cdot 5^{3} \cdot 19^{6} \cdot 41^{6} $
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: $j$  =  \( -\frac{119305480789133569}{192379221760500} \) = $-1 \cdot 2^{-2} \cdot 3^{-4} \cdot 5^{-3} \cdot 7^{3} \cdot 41^{-6} \cdot 70327^{3}$
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: $\mathrm{End}(E)$ = $\Z$
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$  =  \(\Z\)    (no potential complex multiplication)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$
Faltings height: $h_{\mathrm{Faltings}}$ ≈ $2.9041787985085230653154285538$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $1.4319593089253028353109148379$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $1.002573457844343$
Szpiro ratio: $\sigma_{m}$ ≈ $4.484780931237871$
Intrinsic torsion order: $\#E(\mathbb Q)_\text{tors}^\text{is}$ = $1$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 1$
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 1$
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ ≈ $1.7916313045745017576164402296$
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: $\Omega$ ≈ $0.11652276396328396669513251066$
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: $\prod_{p}c_p$ = $ 48 $  = $ 2\cdot2\cdot1\cdot2\cdot( 2 \cdot 3 ) $
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: $\#E(\Q)_{\mathrm{tor}}$ = $2$
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: $ L'(E,1)$ ≈ $2.5051899793459823288583008670 $
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Ш${}_{\mathrm{an}}$  ≈  $1$    (rounded)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

$$\begin{aligned} 2.505189979 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.116523 \cdot 1.791631 \cdot 48}{2^2} \\ & \approx 2.505189979\end{aligned}$$

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([1, 1, 0, -3702423, 5334137433]); 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)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([1, 1, 0, -3702423, 5334137433]); 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 444030.2.a.a

\( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{10} - 6 q^{11} - q^{12} + 4 q^{13} + 4 q^{14} + q^{15} + q^{16} - q^{18} + O(q^{20}) \) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 

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

Modular degree: 49268736
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
Manin constant: 1 (conditional*)
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 
* The Manin constant is correct provided that curve 444030.a3 is optimal.

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2
$3$ $2$ $I_{4}$ nonsplit multiplicative 1 1 4 4
$5$ $1$ $I_{3}$ nonsplit multiplicative 1 1 3 3
$19$ $2$ $I_0^{*}$ additive -1 2 6 0
$41$ $6$ $I_{6}$ split multiplicative -1 1 6 6

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content 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 $\ell$-adic index
$2$ 2B 2.3.0.1 $3$
$3$ 3B 3.4.0.1 $4$

Copy content comment:Mod p Galois image
 
Copy content sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
Copy content magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

Copy content comment:Adelic image of Galois representation
 
Copy content sage:gens = [[46729, 12, 46728, 13], [11, 2, 46690, 46731], [42846, 39577, 42845, 27266], [4919, 0, 0, 46739], [28501, 12312, 13566, 27133], [44290, 14763, 16701, 2452], [1, 0, 12, 1], [1, 6, 6, 37], [1, 12, 0, 1], [15581, 12312, 15580, 1]] GL(2,Integers(46740)).subgroup(gens)
 
Copy content magma:Gens := [[46729, 12, 46728, 13], [11, 2, 46690, 46731], [42846, 39577, 42845, 27266], [4919, 0, 0, 46739], [28501, 12312, 13566, 27133], [44290, 14763, 16701, 2452], [1, 0, 12, 1], [1, 6, 6, 37], [1, 12, 0, 1], [15581, 12312, 15580, 1]]; sub<GL(2,Integers(46740))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 46740 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \cdot 41 \), index $96$, genus $1$, and generators

$\left(\begin{array}{rr} 46729 & 12 \\ 46728 & 13 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 46690 & 46731 \end{array}\right),\left(\begin{array}{rr} 42846 & 39577 \\ 42845 & 27266 \end{array}\right),\left(\begin{array}{rr} 4919 & 0 \\ 0 & 46739 \end{array}\right),\left(\begin{array}{rr} 28501 & 12312 \\ 13566 & 27133 \end{array}\right),\left(\begin{array}{rr} 44290 & 14763 \\ 16701 & 2452 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15581 & 12312 \\ 15580 & 1 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[46740])$ is a degree-$7815633960960000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/46740\Z)$.

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
$2$ nonsplit multiplicative $4$ \( 1805 = 5 \cdot 19^{2} \)
$3$ nonsplit multiplicative $4$ \( 722 = 2 \cdot 19^{2} \)
$5$ nonsplit multiplicative $6$ \( 88806 = 2 \cdot 3 \cdot 19^{2} \cdot 41 \)
$19$ additive $182$ \( 1230 = 2 \cdot 3 \cdot 5 \cdot 41 \)
$41$ split multiplicative $42$ \( 10830 = 2 \cdot 3 \cdot 5 \cdot 19^{2} \)

Isogenies

Copy content comment:Isogenies
 
Copy content gp:ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 444030.a consists of 4 curves linked by isogenies of degrees dividing 6.

Twists

The minimal quadratic twist of this elliptic curve is 1230.h2, its twist by $-19$.

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.