Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2+2376x+352836\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z+2376xz^2+352836z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+3078621x+16415733726\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-22, 550\right)\) | \(\left(27, 648\right)\) |
$\hat{h}(P)$ | ≈ | $0.33341679354231288894151384282$ | $0.69358110901245763824643912278$ |
Integral points
\( \left(-43, 438\right) \), \( \left(-43, -395\right) \), \( \left(-22, 550\right) \), \( \left(-22, -528\right) \), \( \left(0, 594\right) \), \( \left(0, -594\right) \), \( \left(27, 648\right) \), \( \left(27, -675\right) \), \( \left(55, 781\right) \), \( \left(55, -836\right) \), \( \left(143, 1837\right) \), \( \left(143, -1980\right) \), \( \left(162, 2160\right) \), \( \left(162, -2322\right) \), \( \left(272, 4470\right) \), \( \left(272, -4742\right) \), \( \left(594, 14256\right) \), \( \left(594, -14850\right) \), \( \left(2134, 97570\right) \), \( \left(2134, -99704\right) \), \( \left(2673, 136917\right) \), \( \left(2673, -139590\right) \), \( \left(188352, 81649998\right) \), \( \left(188352, -81838350\right) \)
Invariants
Conductor: | \( 54978 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-54337640556276 $ | = | $-1 \cdot 2^{2} \cdot 3^{6} \cdot 7^{7} \cdot 11^{3} \cdot 17 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{12600539783}{461862324} \) | = | $2^{-2} \cdot 3^{-6} \cdot 7^{-1} \cdot 11^{-3} \cdot 13^{3} \cdot 17^{-1} \cdot 179^{3}$ | 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: | $1.3154238436066493871319031555\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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||
Stable Faltings height: | $0.34246876907899273457922678378\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9010252156857624\dots$ | |||
Szpiro ratio: | $3.5791259175353733\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.22870225152756423500900711009\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.47571988740203232774635152139\dots$ | 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: | $ 48 $ = $ 2\cdot2\cdot2^{2}\cdot3\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: | $1$ | 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! $ ≈ $ 5.2223140485736384226968576229 $ | 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
$\displaystyle 5.222314049 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.475720 \cdot 0.228702 \cdot 48}{1^2} \approx 5.222314049$
Modular invariants
Modular form 54978.2.a.b
For more coefficients, see the Downloads section to the right.
Modular degree: | 221184 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$7$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$11$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$17$ | $1$ | $I_{1}$ | 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 |
---|---|---|
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15708 = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 5545 & 6 \\ 927 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 15703 & 6 \\ 15702 & 7 \end{array}\right),\left(\begin{array}{rr} 6731 & 15702 \\ 4485 & 15689 \end{array}\right),\left(\begin{array}{rr} 7855 & 6 \\ 7857 & 19 \end{array}\right),\left(\begin{array}{rr} 7141 & 6 \\ 5715 & 19 \end{array}\right),\left(\begin{array}{rr} 6546 & 9169 \\ 1309 & 2619 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[15708])$ is a degree-$600369109401600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15708\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 54978m
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 7854j1, its twist by $-7$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-7}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.1.5236.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.143548584256.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.2.606414577104.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.0.191909872.1 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.111803582558365659987002374352829551499676840812288.1 | \(\Z/9\Z\) | Not in database |
$18$ | 18.2.29228250191517806613033776870171343410860916736.1 | \(\Z/6\Z\) | 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 | nonsplit | ord | add | split | ord | split | ord | ord | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 4 | 4 | 4 | - | 3 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2,2 |
$\mu$-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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.