Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-1660203x-822916535\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-1660203xz^2-822916535z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-26563251x-52693221490\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-744, 541\right)\) | \(\left(-745, 530\right)\) |
$\hat{h}(P)$ | ≈ | $1.9299660375914009256883422446$ | $3.7431006757618863258226411990$ |
Torsion generators
\( \left(-\frac{2989}{4}, \frac{2989}{8}\right) \)
Integral points
\( \left(-745, 530\right) \), \( \left(-745, 215\right) \), \( \left(-744, 541\right) \), \( \left(-744, 203\right) \), \( \left(1509, 9553\right) \), \( \left(1509, -11062\right) \), \( \left(1622, 26567\right) \), \( \left(1622, -28189\right) \), \( \left(4196, 254951\right) \), \( \left(4196, -259147\right) \), \( \left(6393, 496621\right) \), \( \left(6393, -503014\right) \), \( \left(108092, 35481077\right) \), \( \left(108092, -35589169\right) \)
Invariants
Conductor: | \( 51714 \) | = | $2 \cdot 3^{2} \cdot 13^{2} \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $20501763473815884 $ | = | $2^{2} \cdot 3^{7} \cdot 13^{10} \cdot 17 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{143820170742457}{5826444} \) | = | $2^{-2} \cdot 3^{-1} \cdot 11^{6} \cdot 13^{-4} \cdot 17^{-1} \cdot 433^{3}$ | 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: | $2.2122292702331656805277062265\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.38044844716834246680333988726\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0200658980626969\dots$ | |||
Szpiro ratio: | $5.02888801591161\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $6.3078326900102161984908587024\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.13300265165380337990376613308\dots$ | 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: | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2}\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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 6.7116677916792184326285668814 $ | 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
$\displaystyle 6.711667792 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.133003 \cdot 6.307833 \cdot 32}{2^2} \approx 6.711667792$
Modular invariants
Modular form 51714.2.a.e
For more coefficients, see the Downloads section to the right.
Modular degree: | 688128 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$13$ | $4$ | $I_{4}^{*}$ | Additive | 1 | 2 | 10 | 4 |
$17$ | $1$ | $I_{1}$ | Non-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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5304 = 2^{3} \cdot 3 \cdot 13 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 667 & 666 \\ 2002 & 4651 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 5298 & 5299 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3532 & 5303 \\ 1745 & 5298 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 2039 & 5296 \\ 2852 & 5271 \end{array}\right),\left(\begin{array}{rr} 5297 & 8 \\ 5296 & 9 \end{array}\right),\left(\begin{array}{rr} 1992 & 4649 \\ 2017 & 2064 \end{array}\right),\left(\begin{array}{rr} 2188 & 1 \\ 647 & 6 \end{array}\right)$.
The torsion field $K:=\Q(E[5304])$ is a degree-$3153453907968$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5304\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 51714d
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1326b4, its twist by $-39$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{51}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-221}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-39}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-39}, \sqrt{51})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.2738521903104.78 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | add | ord | ss | ss | add | nonsplit | ss | ss | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 6 | - | 2 | 2,2 | 2,2 | - | 4 | 2,2 | 4,2 | 4 | 2 | 2 | 2 | 2 | 2 |
$\mu$-invariant(s) | 1 | - | 0 | 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.