Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-1003704909x-12239735151735\)
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(homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-1003704909xz^2-12239735151735z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-1300801562739x-571037571215910450\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(1452400630322027/4566650929, 55016681170082943648278/308600569829033)$ | $30.879359283635622946090328634$ | $\infty$ |
$(-18290, 9145)$ | $0$ | $2$ |
$(36582, -18291)$ | $0$ | $2$ |
Integral points
\( \left(-18290, 9145\right) \), \( \left(36582, -18291\right) \)
Invariants
Conductor: | $N$ | = | \( 42978 \) | = | $2 \cdot 3 \cdot 13 \cdot 19 \cdot 29$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $\Delta$ | = | $1532291201797601099556$ | = | $2^{2} \cdot 3^{4} \cdot 13^{2} \cdot 19^{6} \cdot 29^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | $j$ | = | \( \frac{111825759760338976846738658338393}{1532291201797601099556} \) | = | $2^{-2} \cdot 3^{-4} \cdot 7^{3} \cdot 13^{-2} \cdot 19^{-6} \cdot 29^{-6} \cdot 6882547951^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.6208959808317576856472894474$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.6208959808317576856472894474$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $Q$ | ≈ | $1.0459230879697994$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.91708155868041$ |
BSD invariants
Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Mordell-Weil rank: | $r$ | = | $ 1$ | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
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Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $30.879359283635622946090328634$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $\Omega$ | ≈ | $0.026822425204736400204185388345$ | 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: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2\cdot2\cdot2\cdot2\cdot( 2 \cdot 3 ) $ | 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: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ | 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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Special value: | $ L'(E,1)$ | ≈ | $4.9695558285329945048582232237 $ | 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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Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
$\displaystyle 4.969555829 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.026822 \cdot 30.879359 \cdot 96}{4^2} \approx 4.969555829$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 20459520 | 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 at primes of bad reduction
This elliptic curve is semistable. There are 5 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
$3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
$13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
$19$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
$29$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 |
---|---|---|
$2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 57304 = 2^{3} \cdot 13 \cdot 19 \cdot 29 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 45241 & 4 \\ 33178 & 9 \end{array}\right),\left(\begin{array}{rr} 57301 & 4 \\ 57300 & 5 \end{array}\right),\left(\begin{array}{rr} 42981 & 4 \\ 28654 & 3 \end{array}\right),\left(\begin{array}{rr} 4409 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 28653 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 15811 & 2 \\ 27662 & 57303 \end{array}\right)$.
The torsion field $K:=\Q(E[57304])$ is a degree-$70428393249177600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/57304\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | nonsplit multiplicative | $4$ | \( 1 \) |
$3$ | nonsplit multiplicative | $4$ | \( 26 = 2 \cdot 13 \) |
$13$ | split multiplicative | $14$ | \( 3306 = 2 \cdot 3 \cdot 19 \cdot 29 \) |
$19$ | nonsplit multiplicative | $20$ | \( 2262 = 2 \cdot 3 \cdot 13 \cdot 29 \) |
$29$ | split multiplicative | $30$ | \( 1482 = 2 \cdot 3 \cdot 13 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 42978b
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$4$ | \(\Q(\sqrt{13}, \sqrt{-38})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{29}, \sqrt{38})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{-13}, \sqrt{-29})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.2.80951927472.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | ord | ord | split | ord | nonsplit | ss | split | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 2 | 1 | 3 | 1 | 1 | 2 | 1 | 1 | 1,1 | 2 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 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.