Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-244124x+903748061\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-244124xz^2+903748061z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-316384731x+42170015313270\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(431, 29429\right)\) |
$\hat{h}(P)$ | ≈ | $1.4826204323138279040183628211$ |
Torsion generators
\( \left(-\frac{4205}{4}, \frac{4201}{8}\right) \)
Integral points
\( \left(431, 29429\right) \), \( \left(431, -29861\right) \), \( \left(1019, 40895\right) \), \( \left(1019, -41915\right) \)
Invariants
Conductor: | \( 78078 \) | = | $2 \cdot 3 \cdot 7 \cdot 11 \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-351988858573184248416 $ | = | $-1 \cdot 2^{5} \cdot 3^{3} \cdot 7^{8} \cdot 11^{4} \cdot 13^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{333345918055753}{72923718045024} \) | = | $-1 \cdot 2^{-5} \cdot 3^{-3} \cdot 7^{-8} \cdot 11^{-4} \cdot 69337^{3}$ | 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: | $2.6217798338319889009205878590\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.3393051551012205328938441382\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0650939044748338\dots$ | |||
Szpiro ratio: | $4.861539206545962\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.4826204323138279040183628211\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.13893298833300498147744987045\dots$ | 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: | $ 160 $ = $ 5\cdot1\cdot2^{3}\cdot2\cdot2 $ | 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) $ ≈ $ 8.2393954889972741609521333664 $ | 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
$\displaystyle 8.239395489 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.138933 \cdot 1.482620 \cdot 160}{2^2} \approx 8.239395489$
Modular invariants
Modular form 78078.2.a.bz
For more coefficients, see the Downloads section to the right.
Modular degree: | 4423680 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$3$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$7$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$11$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$13$ | $2$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
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 \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 937 & 2912 \\ 52 & 1353 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3426 & 3427 \end{array}\right),\left(\begin{array}{rr} 791 & 0 \\ 0 & 3431 \end{array}\right),\left(\begin{array}{rr} 3425 & 8 \\ 3424 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 92 & 793 \\ 3367 & 534 \end{array}\right),\left(\begin{array}{rr} 235 & 2080 \\ 650 & 1821 \end{array}\right),\left(\begin{array}{rr} 1288 & 2483 \\ 169 & 1782 \end{array}\right)$.
The torsion field $K:=\Q(E[3432])$ is a degree-$531372441600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3432\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 78078.bz
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 462.c3, its twist by $13$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{39}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-26}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-6}, \sqrt{-26})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.87329473560576.41 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.4.1248623850000384.2 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | 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 | split | nonsplit | ord | split | nonsplit | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 4 | 1 | 1 | 2 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 1 | 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.