Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-87674406x+422770263012\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-87674406xz^2+422770263012z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-113626030203x+19725110269178454\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{7}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(5454, 324132\right)\) |
$\hat{h}(P)$ | ≈ | $2.1179145116285031641807299766$ |
Torsion generators
\( \left(4332, 350346\right) \)
Integral points
\( \left(-6684, 846066\right) \), \( \left(-6684, -839382\right) \), \( \left(-3828, 839946\right) \), \( \left(-3828, -836118\right) \), \( \left(3036, 428106\right) \), \( \left(3036, -431142\right) \), \( \left(4332, 350346\right) \), \( \left(4332, -354678\right) \), \( \left(5454, 324132\right) \), \( \left(5454, -329586\right) \), \( \left(14124, 1407882\right) \), \( \left(14124, -1422006\right) \), \( \left(356844, 212915082\right) \), \( \left(356844, -213271926\right) \)
Invariants
Conductor: | \( 64974 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 13 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-34084168945153908664172544 $ | = | $-1 \cdot 2^{21} \cdot 3^{14} \cdot 7^{2} \cdot 13^{2} \cdot 17^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{1521059241134755603512440881}{695595284594977727840256} \) | = | $-1 \cdot 2^{-21} \cdot 3^{-14} \cdot 7 \cdot 13^{-2} \cdot 17^{-7} \cdot 31^{3} \cdot 4159^{3} \cdot 4663^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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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: | $3.6057255219860974347721444591\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $3.2814071638102118839212523352\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0366861869193302\dots$ | |||
Szpiro ratio: | $6.051698981387515\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $2.1179145116285031641807299766\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.061158168660956038507230869549\dots$ | 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: | $ 4116 $ = $ ( 3 \cdot 7 )\cdot( 2 \cdot 7 )\cdot1\cdot2\cdot7 $ | 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)
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Torsion order: | $7$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 10.880332924579636190941687852 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 10.880332925 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.061158 \cdot 2.117915 \cdot 4116}{7^2} \approx 10.880332925$
Modular invariants
Modular form 64974.2.a.bw
For more coefficients, see the Downloads section to the right.
Modular degree: | 14620032 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $21$ | $I_{21}$ | Split multiplicative | -1 | 1 | 21 | 21 |
$3$ | $14$ | $I_{14}$ | Split multiplicative | -1 | 1 | 14 | 14 |
$7$ | $1$ | $II$ | Additive | -1 | 2 | 2 | 0 |
$13$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$17$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
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.1.1 | 7.48.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 952 = 2^{3} \cdot 7 \cdot 17 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 715 & 490 \\ 0 & 171 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 939 & 14 \\ 938 & 15 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 469 & 946 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 231 & 946 \end{array}\right),\left(\begin{array}{rr} 568 & 7 \\ 217 & 946 \end{array}\right)$.
The torsion field $K:=\Q(E[952])$ is a degree-$2526806016$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/952\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 64974bx
consists of 2 curves linked by isogenies of
degree 7.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.6664.1 | \(\Z/14\Z\) | Not in database |
$6$ | 6.0.6039609856.1 | \(\Z/2\Z \oplus \Z/14\Z\) | Not in database |
$8$ | deg 8 | \(\Z/21\Z\) | Not in database |
$12$ | deg 12 | \(\Z/28\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | split | ord | add | ord | nonsplit | split | ord | ord | ord | ord | ord | ss | ord | ord |
$\lambda$-invariant(s) | 4 | 4 | 1 | - | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 |
$\mu$-invariant(s) | 0 | 0 | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.