Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-3008317x-5752578939\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-3008317xz^2-5752578939z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-48133067x-368213185146\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(\frac{9355}{4}, -\frac{9359}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 46970 \) | = | $2 \cdot 5 \cdot 7 \cdot 11 \cdot 61$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-12557142177117000158080 $ | = | $-1 \cdot 2^{7} \cdot 5 \cdot 7 \cdot 11^{16} \cdot 61 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{3010886403566634803117601}{12557142177117000158080} \) | = | $-1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{-16} \cdot 61^{-1} \cdot 48133067^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.9255407567839073125403600575\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.9255407567839073125403600575\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0144868190142529\dots$ | |||
Szpiro ratio: | $5.435325668355168\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.052200802276188282237794766911\dots$ | 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);
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Tamagawa product: | $ 112 $ = $ 7\cdot1\cdot1\cdot2^{4}\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)
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Torsion order: | $2$ | 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: | $4$ = $2^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 5.8464898549330876106330138940 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 5.846489855 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.052201 \cdot 1.000000 \cdot 112}{2^2} \approx 5.846489855$
Modular invariants
Modular form 46970.2.a.o
For more coefficients, see the Downloads section to the right.
Modular degree: | 4444160 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$7$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$11$ | $16$ | $I_{16}$ | Split multiplicative | -1 | 1 | 16 | 16 |
$61$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 |
---|---|---|
$2$ | 2B | 8.12.0.6 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17080 = 2^{3} \cdot 5 \cdot 7 \cdot 61 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 10683 & 10676 \\ 10722 & 2141 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 17074 & 17075 \end{array}\right),\left(\begin{array}{rr} 1968 & 3 \\ 3365 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 4884 & 1 \\ 2463 & 6 \end{array}\right),\left(\begin{array}{rr} 6408 & 14953 \\ 10667 & 10654 \end{array}\right),\left(\begin{array}{rr} 10256 & 3 \\ 5 & 2 \end{array}\right),\left(\begin{array}{rr} 17073 & 8 \\ 17072 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[17080])$ is a degree-$421605015552000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17080\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 46970p
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-4270}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{14}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-305}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{14}, \sqrt{-305})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | 4.2.214251520.3 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | 61 |
---|---|---|---|---|---|---|
Reduction type | split | ss | split | split | split | split |
$\lambda$-invariant(s) | 5 | 0,0 | 1 | 3 | 3 | 1 |
$\mu$-invariant(s) | 2 | 0,0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.