Properties

Label 422370h1
Conductor $422370$
Discriminant $-6.062\times 10^{25}$
j-invariant \( -\frac{480024758533919772826099}{12123360658206720000} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3-x^2-278939115x-1831772148219\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z-278939115xz^2-1831772148219z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-4463025843x-117237880511858\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, -1, 0, -278939115, -1831772148219])
 
gp: E = ellinit([1, -1, 0, -278939115, -1831772148219])
 
magma: E := EllipticCurve([1, -1, 0, -278939115, -1831772148219]);
 
oscar: E = EllipticCurve([1, -1, 0, -278939115, -1831772148219])
 
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

$P$ =  \(\left(70473, 18089526\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $2.9612064280264627621698510540$

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

Integral points

\( \left(70473, 18089526\right) \), \( \left(70473, -18159999\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 422370 \)  =  $2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 19^{2}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-60619361320132481617920000 $  =  $-1 \cdot 2^{15} \cdot 3^{19} \cdot 5^{4} \cdot 13^{5} \cdot 19^{3} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( -\frac{480024758533919772826099}{12123360658206720000} \)  =  $-1 \cdot 2^{-15} \cdot 3^{-13} \cdot 5^{-4} \cdot 13^{-5} \cdot 2609^{3} \cdot 30011^{3}$
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: $3.7313188025760278479464885279\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $2.4459029134503628872466090515\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $2.9612064280264627621698510540\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.018443543943790613296895996521\dots$
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: $ 80 $  = $ 1\cdot2^{2}\cdot2\cdot5\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'(E,1) $ ≈ $ 4.3692112705553041518227540081 $
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

$\displaystyle 4.369211271 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.018444 \cdot 2.961206 \cdot 80}{1^2} \approx 4.369211271$

# 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 422370.2.a.h

\( q - q^{2} + q^{4} - q^{5} - 3 q^{7} - q^{8} + q^{10} - 3 q^{11} + q^{13} + 3 q^{14} + q^{16} + 4 q^{17} + O(q^{20}) \) 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: 169728000
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 5 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $1$ $I_{15}$ Non-split multiplicative 1 1 15 15
$3$ $4$ $I_{13}^{*}$ Additive -1 2 19 13
$5$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$13$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$19$ $2$ $III$ Additive 1 2 3 0

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 = [[5927, 2, 5926, 3], [1, 0, 2, 1], [1, 2, 0, 1], [457, 2, 457, 3], [3953, 2, 3953, 3], [4447, 2, 4447, 3], [1, 1, 5927, 0], [2965, 2, 2965, 3], [3745, 2, 3745, 3]]
 
GL(2,Integers(5928)).subgroup(gens)
 
Gens := [[5927, 2, 5926, 3], [1, 0, 2, 1], [1, 2, 0, 1], [457, 2, 457, 3], [3953, 2, 3953, 3], [4447, 2, 4447, 3], [1, 1, 5927, 0], [2965, 2, 2965, 3], [3745, 2, 3745, 3]];
 
sub<GL(2,Integers(5928))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5928 = 2^{3} \cdot 3 \cdot 13 \cdot 19 \), index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 5927 & 2 \\ 5926 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 457 & 2 \\ 457 & 3 \end{array}\right),\left(\begin{array}{rr} 3953 & 2 \\ 3953 & 3 \end{array}\right),\left(\begin{array}{rr} 4447 & 2 \\ 4447 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 5927 & 0 \end{array}\right),\left(\begin{array}{rr} 2965 & 2 \\ 2965 & 3 \end{array}\right),\left(\begin{array}{rr} 3745 & 2 \\ 3745 & 3 \end{array}\right)$.

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

The torsion field $K:=\Q(E[5928])$ is a degree-$118950136381440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5928\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 422370h consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 140790c1, its twist by $-3$.

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.