Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+701x-1234\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+701xz^2-1234z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+909117x-60289218\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(43, 308\right)\) | \(\left(10, 77\right)\) |
$\hat{h}(P)$ | ≈ | $0.55742520463750191130025533101$ | $1.4300588406394063256507940151$ |
Torsion generators
\( \left(\frac{7}{4}, -\frac{11}{8}\right) \)
Integral points
\( \left(3, 28\right) \), \( \left(3, -32\right) \), \( \left(4, 38\right) \), \( \left(4, -43\right) \), \( \left(10, 77\right) \), \( \left(10, -88\right) \), \( \left(19, 128\right) \), \( \left(19, -148\right) \), \( \left(43, 308\right) \), \( \left(43, -352\right) \), \( \left(88, 818\right) \), \( \left(88, -907\right) \), \( \left(103, 1028\right) \), \( \left(103, -1132\right) \), \( \left(318, 5533\right) \), \( \left(318, -5852\right) \), \( \left(406, 7997\right) \), \( \left(406, -8404\right) \), \( \left(571, 13376\right) \), \( \left(571, -13948\right) \), \( \left(4963, 347168\right) \), \( \left(4963, -352132\right) \), \( \left(16579, 2126432\right) \), \( \left(16579, -2143012\right) \)
Invariants
Conductor: | \( 83490 \) | = | $2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-22812807600 $ | = | $-1 \cdot 2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 11^{3} \cdot 23^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{28680715981}{17139600} \) | = | $2^{-4} \cdot 3^{-4} \cdot 5^{-2} \cdot 23^{-2} \cdot 3061^{3}$ | 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: | $0.67685041433877973204065473429\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.077376596139187096025168839798\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9241365598134934\dots$ | |||
Szpiro ratio: | $2.7596051985865824\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.78931386056213017992531223656\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.70194656934688948180257959847\dots$ | 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: | $ 64 $ = $ 2\cdot2^{2}\cdot2\cdot2\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: | $2$ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 8.8648985049525818751986641336 $ | 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
$\displaystyle 8.864898505 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.701947 \cdot 0.789314 \cdot 64}{2^2} \approx 8.864898505$
Modular invariants
Modular form 83490.2.a.s
For more coefficients, see the Downloads section to the right.
Modular degree: | 86016 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$3$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$11$ | $2$ | $III$ | Additive | 1 | 2 | 3 | 0 |
$23$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5060 = 2^{2} \cdot 5 \cdot 11 \cdot 23 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3797 & 1266 \\ 1264 & 3795 \end{array}\right),\left(\begin{array}{rr} 5057 & 4 \\ 5056 & 5 \end{array}\right),\left(\begin{array}{rr} 3037 & 4 \\ 1014 & 9 \end{array}\right),\left(\begin{array}{rr} 4144 & 1 \\ 2759 & 0 \end{array}\right),\left(\begin{array}{rr} 3961 & 4 \\ 2862 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[5060])$ is a degree-$13542211584000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5060\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 83490.s
consists of 2 curves linked by isogenies of
degree 2.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-11}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.2.281639600.2 | \(\Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | split | nonsplit | ord | add | ord | ord | ord | nonsplit | ss | ord | ord | ss | ord | ord |
$\lambda$-invariant(s) | 2 | 3 | 2 | 2 | - | 2 | 2 | 2 | 2 | 2,2 | 2 | 2 | 2,2 | 2 | 2 |
$\mu$-invariant(s) | 1 | 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.