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Properties

Label 445280c1
Conductor $445280$
Discriminant $-9.914\times 10^{13}$
j-invariant \( -\frac{2318695384896}{1653125} \)
CM no
Rank $3$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-107932x+13656544\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-107932xz^2+13656544z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-107932x+13656544\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 0, 0, -107932, 13656544])
 
gp: E = ellinit([0, 0, 0, -107932, 13656544])
 
magma: E := EllipticCurve([0, 0, 0, -107932, 13656544]);
 
oscar: E = elliptic_curve([0, 0, 0, -107932, 13656544])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generators and heights

$P$ =  \(\left(198, 220\right)\) Copy content Toggle raw display \(\left(\frac{1342}{9}, \frac{25300}{27}\right)\) Copy content Toggle raw display \(\left(\frac{737}{4}, \frac{1265}{8}\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.26005009983764187893832494825$$0.71581928695349978208692283275$$1.2132115840110328839485332539$

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

Integral points

\((-376,\pm 1040)\), \((-362,\pm 2300)\), \((-330,\pm 3652)\), \((-242,\pm 5060)\), \((-187,\pm 5225)\), \((-132,\pm 5060)\), \((-40,\pm 4232)\), \((38,\pm 3100)\), \((88,\pm 2200)\), \((158,\pm 740)\), \((165,\pm 583)\), \((174,\pm 380)\), \((188,\pm 100)\), \((190,\pm 92)\), \((198,\pm 220)\), \((213,\pm 575)\), \((220,\pm 748)\), \((253,\pm 1595)\), \((328,\pm 3680)\), \((374,\pm 5060)\), \((438,\pm 7100)\), \((638,\pm 14300)\), \((828,\pm 22180)\), \((1133,\pm 36685)\), \((1518,\pm 57860)\), \((2398,\pm 116380)\), \((4568,\pm 307960)\), \((8838,\pm 830300)\), \((10373,\pm 1055945)\), \((11110,\pm 1170532)\), \((35288,\pm 6628600)\), \((35398,\pm 6659620)\), \((9104958,\pm 27473685340)\), \((14593052518,\pm 1762865660789860)\) Copy content Toggle raw display

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

Invariants

Conductor: \( 445280 \)  =  $2^{5} \cdot 5 \cdot 11^{2} \cdot 23$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-99137139200000 $  =  $-1 \cdot 2^{12} \cdot 5^{5} \cdot 11^{4} \cdot 23^{2} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( -\frac{2318695384896}{1653125} \)  =  $-1 \cdot 2^{6} \cdot 3^{3} \cdot 5^{-5} \cdot 11^{2} \cdot 23^{-2} \cdot 223^{3}$
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: $1.6212066588355656880096490781\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.12876105400949686390510243065\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $0.9337438013959558\dots$
Szpiro ratio: $3.566120940695633\dots$

BSD invariants

Analytic rank: $3$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $0.20582790600993815034179936646\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.59334098804149459230685602273\dots$
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: $ 120 $  = $ 2^{2}\cdot5\cdot3\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: $1$
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^{(3)}(E,1)/3! $ ≈ $ 14.655135974213830198554244547 $
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

$\displaystyle 14.655135974 \approx L^{(3)}(E,1)/3! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.593341 \cdot 0.205828 \cdot 120}{1^2} \approx 14.655135974$

# 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 445280.2.a.c

\( q - 3 q^{3} + q^{5} - 3 q^{7} + 6 q^{9} - 6 q^{13} - 3 q^{15} - 6 q^{17} - 6 q^{19} + O(q^{20}) \) 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: 4792320
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 $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $4$ $I_{3}^{*}$ additive -1 5 12 0
$5$ $5$ $I_{5}$ split multiplicative -1 1 5 5
$11$ $3$ $IV$ additive -1 2 4 0
$23$ $2$ $I_{2}$ split 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 $\ell$.

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 = [[1, 2, 0, 1], [1, 1, 19, 0], [17, 2, 17, 3], [1, 0, 2, 1], [11, 2, 11, 3], [19, 2, 18, 3]]
 
GL(2,Integers(20)).subgroup(gens)
 
Gens := [[1, 2, 0, 1], [1, 1, 19, 0], [17, 2, 17, 3], [1, 0, 2, 1], [11, 2, 11, 3], [19, 2, 18, 3]];
 
sub<GL(2,Integers(20))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 20.2.0.a.1, level \( 20 = 2^{2} \cdot 5 \), index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 19 & 0 \end{array}\right),\left(\begin{array}{rr} 17 & 2 \\ 17 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 11 & 3 \end{array}\right),\left(\begin{array}{rr} 19 & 2 \\ 18 & 3 \end{array}\right)$.

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

The torsion field $K:=\Q(E[20])$ is a degree-$23040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/20\Z)$.

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
$2$ additive $2$ \( 605 = 5 \cdot 11^{2} \)
$5$ split multiplicative $6$ \( 89056 = 2^{5} \cdot 11^{2} \cdot 23 \)
$11$ additive $52$ \( 3680 = 2^{5} \cdot 5 \cdot 23 \)
$23$ split multiplicative $24$ \( 19360 = 2^{5} \cdot 5 \cdot 11^{2} \)

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 445280c consists of this curve only.

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.