Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-498774543x-4287535531454\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-498774543xz^2-4287535531454z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-646411807107x-200037318520084866\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-\frac{11004830003}{853776}, \frac{2697025300117}{788889024}\right)\) |
$\hat{h}(P)$ | ≈ | $18.108593324196740606518272353$ |
Torsion generators
\( \left(-\frac{51641}{4}, \frac{51637}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 62790 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $36939806611960382108160 $ | = | $2^{9} \cdot 3 \cdot 5 \cdot 7^{12} \cdot 13^{4} \cdot 23^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{13722604572968640187892492722921}{36939806611960382108160} \) | = | $2^{-9} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-12} \cdot 11^{3} \cdot 13^{-4} \cdot 23^{-3} \cdot 229^{3} \cdot 9504239^{3}$ | 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: | $3.5646310909852533879682452813\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $3.5646310909852533879682452813\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0127394271927954\dots$ | |||
Szpiro ratio: | $6.4898181104698365\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $18.108593324196740606518272353\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.031946519897132968313985917388\dots$ | 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: | $ 48 $ = $ 1\cdot1\cdot1\cdot( 2^{2} \cdot 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)
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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'(E,1) $ ≈ $ 6.9420784432864849734199069841 $ | 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
$\displaystyle 6.942078443 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.031947 \cdot 18.108593 \cdot 48}{2^2} \approx 6.942078443$
Modular invariants
Modular form 62790.2.a.v
For more coefficients, see the Downloads section to the right.
Modular degree: | 19906560 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{9}$ | Non-split multiplicative | 1 | 1 | 9 | 9 |
$3$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$7$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$13$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$23$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
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 |
$3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 251160 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 23 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 251137 & 24 \\ 251136 & 25 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 249854 & 241931 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 62791 & 24 \\ 62802 & 289 \end{array}\right),\left(\begin{array}{rr} 150712 & 3 \\ 150237 & 251074 \end{array}\right),\left(\begin{array}{rr} 52326 & 10489 \\ 115115 & 73256 \end{array}\right),\left(\begin{array}{rr} 215281 & 24 \\ 71772 & 289 \end{array}\right),\left(\begin{array}{rr} 10936 & 21 \\ 229035 & 250786 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 135241 & 24 \\ 115932 & 289 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 83736 & 41873 \\ 126311 & 594 \end{array}\right)$.
The torsion field $K:=\Q(E[251160])$ is a degree-$1300922165247344640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/251160\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 62790.v
consists of 8 curves linked by isogenies of
degrees dividing 12.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{690}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-69}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-10}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/6\Z\) | Not in database |
$3$ | 3.1.1026675.4 | \(\Z/6\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-10}, \sqrt{-69})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{-230})\) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{23})\) | \(\Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{-10})\) | \(\Z/12\Z\) | Not in database |
$6$ | 6.0.3162184666875.2 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.2.98494210155182400000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.0.2462355253879560000.1 | \(\Z/12\Z\) | Not in database |
$6$ | 6.0.2698397582400000.2 | \(\Z/12\Z\) | 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 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.928445276160000.143 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$18$ | 18.0.59700265403687827410764105372567068414202359258604296875.1 | \(\Z/18\Z\) | 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 | split | split | ss | split | ord | ord | nonsplit | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 5 | 2 | 4 | 4 | 1,1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1,3 |
$\mu$-invariant(s) | 1 | 1 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.