Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-265716552x+1667217712779\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-265716552xz^2+1667217712779z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4251464827x+106697682153046\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{7}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(9337, 7681\right) \) | $2.2570873815132427145303379356$ | $\infty$ |
| \( \left(9437, -919\right) \) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([9337:7681:1]\) | $2.2570873815132427145303379356$ | $\infty$ |
| \([9437:-919:1]\) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(37347, 98800\right) \) | $2.2570873815132427145303379356$ | $\infty$ |
| \( \left(37747, 30400\right) \) | $0$ | $7$ |
Integral points
\( \left(-11463, 1796481\right) \), \( \left(-11463, -1785019\right) \), \( \left(7309, 336217\right) \), \( \left(7309, -343527\right) \), \( \left(9337, 7681\right) \), \( \left(9337, -17019\right) \), \( \left(9437, -919\right) \), \( \left(9437, -8519\right) \), \( \left(9817, 63681\right) \), \( \left(9817, -73499\right) \), \( \left(17037, 1435481\right) \), \( \left(17037, -1452519\right) \)
\([-11463:1796481:1]\), \([-11463:-1785019:1]\), \([7309:336217:1]\), \([7309:-343527:1]\), \([9337:7681:1]\), \([9337:-17019:1]\), \([9437:-919:1]\), \([9437:-8519:1]\), \([9817:63681:1]\), \([9817:-73499:1]\), \([17037:1435481:1]\), \([17037:-1452519:1]\)
\((-45853,\pm 14326000)\), \((29235,\pm 2718976)\), \((37347,\pm 98800)\), \((37747,\pm 30400)\), \((39267,\pm 548720)\), \((68147,\pm 11552000)\)
Invariants
| Conductor: | $N$ | = | \( 71630 \) | = | $2 \cdot 5 \cdot 13 \cdot 19 \cdot 29$ |
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| Minimal Discriminant: | $\Delta$ | = | $5607507702833920000000$ | = | $2^{14} \cdot 5^{7} \cdot 13^{2} \cdot 19^{7} \cdot 29 $ |
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| j-invariant: | $j$ | = | \( \frac{2074815747201021028933709986641}{5607507702833920000000} \) | = | $2^{-14} \cdot 3^{3} \cdot 5^{-7} \cdot 13^{-2} \cdot 19^{-7} \cdot 29^{-1} \cdot 127^{3} \cdot 33476101^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.4073011602687455781638744244$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.4073011602687455781638744244$ |
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| $abc$ quality: | $Q$ | ≈ | $1.006892666415928$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.24436387763295$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.2570873815132427145303379356$ |
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| Real period: | $\Omega$ | ≈ | $0.11741425693870218870553452208$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1372 $ = $ ( 2 \cdot 7 )\cdot7\cdot2\cdot7\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $7$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.4203986568907555528034936229 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.420398657 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.117414 \cdot 2.257087 \cdot 1372}{7^2} \\ & \approx 7.420398657\end{aligned}$$
Modular invariants
Modular form 71630.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 26254592 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $14$ | $I_{14}$ | split multiplicative | -1 | 1 | 14 | 14 |
| $5$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $29$ | $1$ | $I_{1}$ | 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 | $\ell$-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 \( 77140 = 2^{2} \cdot 5 \cdot 7 \cdot 19 \cdot 29 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 8 & 7 \\ 38563 & 77134 \end{array}\right),\left(\begin{array}{rr} 77127 & 14 \\ 77126 & 15 \end{array}\right),\left(\begin{array}{rr} 15436 & 7 \\ 15421 & 77134 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 74488 & 7 \\ 50533 & 77134 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 20308 & 7 \\ 69013 & 77134 \end{array}\right),\left(\begin{array}{rr} 38571 & 14 \\ 0 & 49591 \end{array}\right)$.
The torsion field $K:=\Q(E[77140])$ is a degree-$81263530672128000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/77140\Z)$.
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 | $4$ | \( 2755 = 5 \cdot 19 \cdot 29 \) |
| $5$ | split multiplicative | $6$ | \( 14326 = 2 \cdot 13 \cdot 19 \cdot 29 \) |
| $7$ | good | $2$ | \( 377 = 13 \cdot 29 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 5510 = 2 \cdot 5 \cdot 19 \cdot 29 \) |
| $19$ | split multiplicative | $20$ | \( 3770 = 2 \cdot 5 \cdot 13 \cdot 29 \) |
| $29$ | split multiplicative | $30$ | \( 2470 = 2 \cdot 5 \cdot 13 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 71630x
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 |
|---|---|---|---|
| $3$ | 3.3.11020.1 | \(\Z/14\Z\) | not in database |
| $6$ | 6.6.1338273208000.1 | \(\Z/2\Z \oplus \Z/14\Z\) | 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 | ss | split | ord | ord | nonsplit | ord | split | ord | split | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 1,1 | 4 | 51 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 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.