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Properties

Label 442090b1
Conductor $442090$
Discriminant $-1.642\times 10^{24}$
j-invariant \( \frac{3520454064209678324329119}{1642467221810708480000000} \)
CM no
Rank $1$
Torsion structure \(\Z/{7}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-x^2+3169263x+61621374849\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-x^2z+3169263xz^2+61621374849z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+50708213x+3943818698566\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, -1, 1, 3169263, 61621374849])
 
Copy content gp:E = ellinit([1, -1, 1, 3169263, 61621374849])
 
Copy content magma:E := EllipticCurve([1, -1, 1, 3169263, 61621374849]);
 
Copy content oscar:E = elliptic_curve([1, -1, 1, 3169263, 61621374849])
 
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

\(\Z \oplus \Z/{7}\Z\)

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
\( \left(847, 254356\right) \)$5.5222570838681468207120719616$$\infty$
\( \left(1607, 265396\right) \)$0$$7$

$P$$\hat{h}(P)$Order
\([847:254356:1]\)$5.5222570838681468207120719616$$\infty$
\([1607:265396:1]\)$0$$7$

$P$$\hat{h}(P)$Order
\( \left(3387, 2038240\right) \)$5.5222570838681468207120719616$$\infty$
\( \left(6427, 2129600\right) \)$0$$7$

Integral points

\( \left(-2793, 177396\right) \), \( \left(-2793, -174604\right) \), \( \left(847, 254356\right) \), \( \left(847, -255204\right) \), \( \left(1607, 265396\right) \), \( \left(1607, -267004\right) \), \( \left(11287, 1233396\right) \), \( \left(11287, -1244684\right) \), \( \left(467457, 319372646\right) \), \( \left(467457, -319840104\right) \) Copy content Toggle raw display

\([-2793:177396:1]\), \([-2793:-174604:1]\), \([847:254356:1]\), \([847:-255204:1]\), \([1607:265396:1]\), \([1607:-267004:1]\), \([11287:1233396:1]\), \([11287:-1244684:1]\), \([467457:319372646:1]\), \([467457:-319840104:1]\) Copy content Toggle raw display

\((-11173,\pm 1408000)\), \((3387,\pm 2038240)\), \((6427,\pm 2129600)\), \((45147,\pm 9912320)\), \((1869827,\pm 2556851000)\) Copy content Toggle raw display

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

Invariants

Conductor: $N$  =  \( 442090 \) = $2 \cdot 5 \cdot 11 \cdot 4019$
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: $\Delta$  =  $-1642467221810708480000000$ = $-1 \cdot 2^{28} \cdot 5^{7} \cdot 11^{7} \cdot 4019 $
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: $j$  =  \( \frac{3520454064209678324329119}{1642467221810708480000000} \) = $2^{-28} \cdot 3^{3} \cdot 5^{-7} \cdot 11^{-7} \cdot 67^{3} \cdot 4019^{-1} \cdot 756839^{3}$
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: $h_{\mathrm{Faltings}}$ ≈ $3.3256065165222728426341471185$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $3.3256065165222728426341471185$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $0.9961366585378372$
Szpiro ratio: $\sigma_{m}$ ≈ $4.862713565058383$
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: $\mathrm{Reg}(E/\Q)$ ≈ $5.5222570838681468207120719616$
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: $\Omega$ ≈ $0.065533088185268468979099441811$
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: $\prod_{p}c_p$ = $ 1372 $  = $ ( 2^{2} \cdot 7 )\cdot7\cdot7\cdot1 $
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: $\#E(\Q)_{\mathrm{tor}}$ = $7$
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)$ ≈ $10.132935692847933314604705474 $
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

$$\begin{aligned} 10.132935693 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.065533 \cdot 5.522257 \cdot 1372}{7^2} \\ & \approx 10.132935693\end{aligned}$$

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, 1, 3169263, 61621374849]); 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, 1, 3169263, 61621374849]); 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 442090.2.a.b

\( q + q^{2} - 3 q^{3} + q^{4} + q^{5} - 3 q^{6} + q^{7} + q^{8} + 6 q^{9} + q^{10} + q^{11} - 3 q^{12} + q^{14} - 3 q^{15} + q^{16} + 4 q^{17} + 6 q^{18} - q^{19} + O(q^{20}) \) 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: 116169984
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: yes
Manin constant: 1
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is semistable. There are 4 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $28$ $I_{28}$ split multiplicative -1 1 28 28
$5$ $7$ $I_{7}$ split multiplicative -1 1 7 7
$11$ $7$ $I_{7}$ split multiplicative -1 1 7 7
$4019$ $1$ $I_{1}$ split multiplicative -1 1 1 1

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
$7$ 7B.1.1 7.48.0.1 $48$

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 = [[3094631, 14, 3094637, 99], [1, 14, 0, 1], [5626601, 14, 2250647, 99], [164781, 14, 1153467, 99], [3094631, 14, 0, 5747171], [1, 0, 14, 1], [8, 5, 91, 57], [6189247, 14, 6189246, 15], [3713557, 14, 1237859, 99]] GL(2,Integers(6189260)).subgroup(gens)
 
Copy content magma:Gens := [[3094631, 14, 3094637, 99], [1, 14, 0, 1], [5626601, 14, 2250647, 99], [164781, 14, 1153467, 99], [3094631, 14, 0, 5747171], [1, 0, 14, 1], [8, 5, 91, 57], [6189247, 14, 6189246, 15], [3713557, 14, 1237859, 99]]; sub<GL(2,Integers(6189260))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6189260 = 2^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 4019 \), index $96$, genus $2$, and generators

$\left(\begin{array}{rr} 3094631 & 14 \\ 3094637 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5626601 & 14 \\ 2250647 & 99 \end{array}\right),\left(\begin{array}{rr} 164781 & 14 \\ 1153467 & 99 \end{array}\right),\left(\begin{array}{rr} 3094631 & 14 \\ 0 & 5747171 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 6189247 & 14 \\ 6189246 & 15 \end{array}\right),\left(\begin{array}{rr} 3713557 & 14 \\ 1237859 & 99 \end{array}\right)$.

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

The torsion field $K:=\Q(E[6189260])$ is a degree-$3331728627375037317120000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6189260\Z)$.

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 $4$ \( 221045 = 5 \cdot 11 \cdot 4019 \)
$5$ split multiplicative $6$ \( 88418 = 2 \cdot 11 \cdot 4019 \)
$7$ good $2$ \( 4019 \)
$11$ split multiplicative $12$ \( 40190 = 2 \cdot 5 \cdot 4019 \)
$4019$ split multiplicative $4020$ \( 110 = 2 \cdot 5 \cdot 11 \)

Isogenies

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

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 442090b consists of 2 curves linked by isogenies of degree 7.

Twists

This elliptic curve is its own minimal quadratic twist.

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.