Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-372654x+87584328\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-372654xz^2+87584328z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-5962467x+5599434526\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(387, 909\right)\) | \(\left(142, 6054\right)\) |
$\hat{h}(P)$ | ≈ | $0.48192705758148414832154019743$ | $0.64104342363246239131902651352$ |
Torsion generators
\( \left(\frac{1443}{4}, -\frac{1443}{8}\right) \)
Integral points
\( \left(-698, 3044\right) \), \( \left(-698, -2346\right) \), \( \left(-666, 6660\right) \), \( \left(-666, -5994\right) \), \( \left(-558, 11304\right) \), \( \left(-558, -10746\right) \), \( \left(-348, 13404\right) \), \( \left(-348, -13056\right) \), \( \left(-108, 11304\right) \), \( \left(-108, -11196\right) \), \( \left(142, 6054\right) \), \( \left(142, -6196\right) \), \( \left(234, 3510\right) \), \( \left(234, -3744\right) \), \( \left(282, 2064\right) \), \( \left(282, -2346\right) \), \( \left(324, 720\right) \), \( \left(324, -1044\right) \), \( \left(342, 54\right) \), \( \left(342, -396\right) \), \( \left(367, 204\right) \), \( \left(367, -571\right) \), \( \left(373, 426\right) \), \( \left(373, -799\right) \), \( \left(387, 909\right) \), \( \left(387, -1296\right) \), \( \left(417, 1929\right) \), \( \left(417, -2346\right) \), \( \left(667, 11304\right) \), \( \left(667, -11971\right) \), \( \left(1017, 27054\right) \), \( \left(1017, -28071\right) \), \( \left(1767, 69429\right) \), \( \left(1767, -71196\right) \), \( \left(3117, 169329\right) \), \( \left(3117, -172446\right) \), \( \left(4797, 327249\right) \), \( \left(4797, -332046\right) \), \( \left(14247, 1691829\right) \), \( \left(14247, -1706076\right) \)
Invariants
Conductor: | \( 57330 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $5226372998437500 $ | = | $2^{2} \cdot 3^{7} \cdot 5^{8} \cdot 7^{6} \cdot 13 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{66730743078481}{60937500} \) | = | $2^{-2} \cdot 3^{-1} \cdot 5^{-8} \cdot 13^{-1} \cdot 47^{3} \cdot 863^{3}$ | 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: | $1.9397609001809143273840551861\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.41749968131920282913375619592\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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||
$abc$ quality: | $0.9796848210169614\dots$ | |||
Szpiro ratio: | $4.572487453842685\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.27922706644606080182242419196\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.42766944056288643142701505671\dots$ | 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: | $ 256 $ = $ 2\cdot2^{2}\cdot2^{3}\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)
|
Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
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! $ ≈ $ 7.6426805310081753799042760946 $ | 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
$\displaystyle 7.642680531 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.427669 \cdot 0.279227 \cdot 256}{2^2} \approx 7.642680531$
Modular invariants
Modular form 57330.2.a.bu
For more coefficients, see the Downloads section to the right.
Modular degree: | 786432 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$5$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$7$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$13$ | $1$ | $I_{1}$ | 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 | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 21840 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 8737 & 3136 \\ 16856 & 3249 \end{array}\right),\left(\begin{array}{rr} 14176 & 18725 \\ 2835 & 9346 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21825 & 16 \\ 21824 & 17 \end{array}\right),\left(\begin{array}{rr} 3119 & 0 \\ 0 & 21839 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 21742 & 21827 \end{array}\right),\left(\begin{array}{rr} 14552 & 9359 \\ 4081 & 18710 \end{array}\right),\left(\begin{array}{rr} 18733 & 3136 \\ 2604 & 8905 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 13658 & 791 \\ 20587 & 17046 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 21836 & 21837 \end{array}\right)$.
The torsion field $K:=\Q(E[21840])$ is a degree-$155817722511360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/21840\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 57330.bu
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 390.f2, its twist by $21$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{39}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{21}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{91}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{21}, \sqrt{39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{21}, \sqrt{26})\) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{6}, \sqrt{14})\) | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.2162816965161216.46 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | 8.8.364024420171776.12 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | \(\Z/16\Z\) | 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 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | add | split | add | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 6 | - | 3 | - | 2 | 2 | 2 | 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 |
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.