Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-459769306x+3796241996132\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-459769306xz^2+3796241996132z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-595861020603x+177119254154596374\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{7}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(9812, 474326\right) \) | $2.5752300457588896697948367701$ | $\infty$ |
| \( \left(12116, 59606\right) \) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([9812:474326:1]\) | $2.5752300457588896697948367701$ | $\infty$ |
| \([12116:59606:1]\) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(353235, 103514112\right) \) | $2.5752300457588896697948367701$ | $\infty$ |
| \( \left(436179, 14183424\right) \) | $0$ | $7$ |
Integral points
\( \left(-2476, 2219222\right) \), \( \left(-2476, -2216746\right) \), \( \left(9812, 474326\right) \), \( \left(9812, -484138\right) \), \( \left(10292, 387926\right) \), \( \left(10292, -398218\right) \), \( \left(12116, 59606\right) \), \( \left(12116, -71722\right) \), \( \left(12686, 66218\right) \), \( \left(12686, -78904\right) \), \( \left(14852, 486422\right) \), \( \left(14852, -501274\right) \), \( \left(34004, 5225174\right) \), \( \left(34004, -5259178\right) \), \( \left(40844, 7270334\right) \), \( \left(40844, -7311178\right) \)
\([-2476:2219222:1]\), \([-2476:-2216746:1]\), \([9812:474326:1]\), \([9812:-484138:1]\), \([10292:387926:1]\), \([10292:-398218:1]\), \([12116:59606:1]\), \([12116:-71722:1]\), \([12686:66218:1]\), \([12686:-78904:1]\), \([14852:486422:1]\), \([14852:-501274:1]\), \([34004:5225174:1]\), \([34004:-5259178:1]\), \([40844:7270334:1]\), \([40844:-7311178:1]\)
\((-89133,\pm 479084544)\), \((353235,\pm 103514112)\), \((370515,\pm 84903552)\), \((436179,\pm 14183424)\), \((456699,\pm 15673176)\), \((534675,\pm 106671168)\), \((1224147,\pm 1132310016)\), \((1470387,\pm 1574803296)\)
Invariants
| Conductor: | $N$ | = | \( 42978 \) | = | $2 \cdot 3 \cdot 13 \cdot 19 \cdot 29$ |
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| Minimal Discriminant: | $\Delta$ | = | $-5737242625602477531070464$ | = | $-1 \cdot 2^{28} \cdot 3^{7} \cdot 13 \cdot 19^{7} \cdot 29^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{10748395438529140294639078020769}{5737242625602477531070464} \) | = | $-1 \cdot 2^{-28} \cdot 3^{-7} \cdot 13^{-1} \cdot 19^{-7} \cdot 29^{-2} \cdot 83^{3} \cdot 313^{3} \cdot 547^{3} \cdot 1553^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.7003239301159229423821075166$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.7003239301159229423821075166$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0176711500540818$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.69762466313401$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5752300457588896697948367701$ |
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| Real period: | $\Omega$ | ≈ | $0.074946059990541385505990626184$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2744 $ = $ ( 2^{2} \cdot 7 )\cdot7\cdot1\cdot7\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $7$ |
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| Special value: | $ L'(E,1)$ | ≈ | $10.808187347937861419417631778 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.808187348 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.074946 \cdot 2.575230 \cdot 2744}{7^2} \\ & \approx 10.808187348\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 11063808 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $28$ | $I_{28}$ | split multiplicative | -1 | 1 | 28 | 28 |
| $3$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $29$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | $\ell$-adic index |
|---|---|---|---|
| $7$ | 7B.1.1 | 7.48.0.1 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 20748 = 2^{2} \cdot 3 \cdot 7 \cdot 13 \cdot 19 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 18565 & 14 \\ 5467 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10375 & 14 \\ 10381 & 99 \end{array}\right),\left(\begin{array}{rr} 20735 & 14 \\ 20734 & 15 \end{array}\right),\left(\begin{array}{rr} 10375 & 14 \\ 0 & 19267 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 6385 & 14 \\ 3199 & 99 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 6917 & 14 \\ 6923 & 99 \end{array}\right)$.
The torsion field $K:=\Q(E[20748])$ is a degree-$312244108001280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/20748\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 741 = 3 \cdot 13 \cdot 19 \) |
| $3$ | split multiplicative | $4$ | \( 14326 = 2 \cdot 13 \cdot 19 \cdot 29 \) |
| $7$ | good | $2$ | \( 377 = 13 \cdot 29 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 3306 = 2 \cdot 3 \cdot 19 \cdot 29 \) |
| $19$ | split 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 $d=$
7.
Its isogeny class 42978r
consists of 2 curves linked by isogenies of
degree 7.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.2964.1 | \(\Z/14\Z\) | not in database |
| $6$ | 6.0.26039617344.2 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $8$ | deg 8 | \(\Z/21\Z\) | not in database |
| $12$ | deg 12 | \(\Z/28\Z\) | 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 | split | ord | ord | ord | nonsplit | ord | split | ord | split | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 4 | 2 | 7 | 51 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1,1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.