Properties

Label 433698q1
Conductor $433698$
Discriminant $-7.319\times 10^{18}$
j-invariant \( \frac{444369620591}{1540767744} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3+x^2+267244x+118914005\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3+x^2z+267244xz^2+118914005z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+346348197x+5542856602614\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 1, 1, 267244, 118914005])
 
gp: E = ellinit([1, 1, 1, 267244, 118914005])
 
magma: E := EllipticCurve([1, 1, 1, 267244, 118914005]);
 
oscar: E = EllipticCurve([1, 1, 1, 267244, 118914005])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generator and height

$P$ =  \(\left(1069, 39809\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $1.6222077237204861135145996199$

sage: E.gens()
 
magma: Generators(E);
 
gp: E.gen
 

Integral points

\( \left(1069, 39809\right) \), \( \left(1069, -40879\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 433698 \)  =  $2 \cdot 3 \cdot 41^{2} \cdot 43$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-7318807395170402304 $  =  $-1 \cdot 2^{14} \cdot 3^{7} \cdot 41^{6} \cdot 43 $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{444369620591}{1540767744} \)  =  $2^{-14} \cdot 3^{-7} \cdot 13^{3} \cdot 43^{-1} \cdot 587^{3}$
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.3027529564830366208290193256\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.44596692313088271889563763908\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $1.6222077237204861135145996199\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.16680671868584795624849890486\dots$
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: $ 28 $  = $ ( 2 \cdot 7 )\cdot1\cdot2\cdot1 $
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: $1$
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Analytic order of Ш: $1$ ( rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L'(E,1) $ ≈ $ 7.5766641277782809167567814691 $
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);
 

BSD formula

$\displaystyle 7.576664128 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.166807 \cdot 1.622208 \cdot 28}{1^2} \approx 7.576664128$

# 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 analyiic 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 433698.2.a.q

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

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: 11854080
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $14$ $I_{14}$ Split multiplicative -1 1 14 14
$3$ $1$ $I_{7}$ Non-split multiplicative 1 1 7 7
$41$ $2$ $I_0^{*}$ Additive 1 2 6 0
$43$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

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
$7$ 7B.6.1 7.24.0.1

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 = [[74047, 61418, 0, 95203], [104756, 104755, 117383, 43338], [1, 14, 0, 1], [6028, 104755, 92701, 43338], [118532, 104755, 105329, 43338], [1, 0, 14, 1], [8, 5, 91, 57], [148079, 14, 148078, 15], [32507, 0, 0, 148091]]
 
GL(2,Integers(148092)).subgroup(gens)
 
Gens := [[74047, 61418, 0, 95203], [104756, 104755, 117383, 43338], [1, 14, 0, 1], [6028, 104755, 92701, 43338], [118532, 104755, 105329, 43338], [1, 0, 14, 1], [8, 5, 91, 57], [148079, 14, 148078, 15], [32507, 0, 0, 148091]];
 
sub<GL(2,Integers(148092))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 148092 = 2^{2} \cdot 3 \cdot 7 \cdot 41 \cdot 43 \), index $96$, genus $2$, and generators

$\left(\begin{array}{rr} 74047 & 61418 \\ 0 & 95203 \end{array}\right),\left(\begin{array}{rr} 104756 & 104755 \\ 117383 & 43338 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6028 & 104755 \\ 92701 & 43338 \end{array}\right),\left(\begin{array}{rr} 118532 & 104755 \\ 105329 & 43338 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 148079 & 14 \\ 148078 & 15 \end{array}\right),\left(\begin{array}{rr} 32507 & 0 \\ 0 & 148091 \end{array}\right)$.

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

The torsion field $K:=\Q(E[148092])$ is a degree-$889825009257676800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/148092\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 433698q consists of 2 curves linked by isogenies of degree 7.

Twists

The minimal quadratic twist of this elliptic curve is 258f1, its twist by $41$.

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.