Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-35874060x+82698195600\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-35874060xz^2+82698195600z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-46492781787x+3858506492258934\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{7}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3470, -1360\right) \) | $2.7690315184735211907602513638$ | $\infty$ |
| \( \left(3480, 180\right) \) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([3470:-1360:1]\) | $2.7690315184735211907602513638$ | $\infty$ |
| \([3480:180:1]\) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(124923, 81000\right) \) | $2.7690315184735211907602513638$ | $\infty$ |
| \( \left(125283, 414720\right) \) | $0$ | $7$ |
Integral points
\( \left(3470, -1360\right) \), \( \left(3470, -2110\right) \), \( \left(3480, 180\right) \), \( \left(3480, -3660\right) \), \( \left(3720, 25140\right) \), \( \left(3720, -28860\right) \), \( \left(4152, 70932\right) \), \( \left(4152, -75084\right) \), \( \left(4800, 142860\right) \), \( \left(4800, -147660\right) \), \( \left(7320, 457140\right) \), \( \left(7320, -464460\right) \), \( \left(16920, 2069940\right) \), \( \left(16920, -2086860\right) \), \( \left(45720, 9673140\right) \), \( \left(45720, -9718860\right) \)
\([3470:-1360:1]\), \([3470:-2110:1]\), \([3480:180:1]\), \([3480:-3660:1]\), \([3720:25140:1]\), \([3720:-28860:1]\), \([4152:70932:1]\), \([4152:-75084:1]\), \([4800:142860:1]\), \([4800:-147660:1]\), \([7320:457140:1]\), \([7320:-464460:1]\), \([16920:2069940:1]\), \([16920:-2086860:1]\), \([45720:9673140:1]\), \([45720:-9718860:1]\)
\((124923,\pm 81000)\), \((125283,\pm 414720)\), \((133923,\pm 5832000)\), \((149475,\pm 15769728)\), \((172803,\pm 31376160)\), \((263523,\pm 99532800)\), \((609123,\pm 448934400)\), \((1645923,\pm 2094336000)\)
Invariants
| Conductor: | $N$ | = | \( 63870 \) | = | $2 \cdot 3 \cdot 5 \cdot 2129$ |
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| Minimal Discriminant: | $\Delta$ | = | $97645976616960000000$ | = | $2^{28} \cdot 3^{7} \cdot 5^{7} \cdot 2129 $ |
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| j-invariant: | $j$ | = | \( \frac{5105817686570071165887579841}{97645976616960000000} \) | = | $2^{-28} \cdot 3^{-7} \cdot 5^{-7} \cdot 241^{3} \cdot 2129^{-1} \cdot 7145041^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9589217535955948931421486178$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.9589217535955948931421486178$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9932863758556641$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.766150518238079$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.7690315184735211907602513638$ |
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| Real period: | $\Omega$ | ≈ | $0.17448909532839870654139304699$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1372 $ = $ ( 2^{2} \cdot 7 )\cdot7\cdot7\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $7$ |
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| Special value: | $ L'(E,1)$ | ≈ | $13.528642528639472166187709657 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.528642529 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.174489 \cdot 2.769032 \cdot 1372}{7^2} \\ & \approx 13.528642529\end{aligned}$$
Modular invariants
Modular form 63870.2.a.o
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4346496 |
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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 4 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 |
| $5$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $2129$ | $1$ | $I_{1}$ | 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 \( 894180 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 2129 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 447091 & 14 \\ 0 & 63871 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 800528 & 7 \\ 864353 & 894174 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 536516 & 7 \\ 536501 & 894174 \end{array}\right),\left(\begin{array}{rr} 894167 & 14 \\ 894166 & 15 \end{array}\right),\left(\begin{array}{rr} 596128 & 7 \\ 596113 & 894174 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 447083 & 894174 \end{array}\right)$.
The torsion field $K:=\Q(E[894180])$ is a degree-$953831179052856115200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/894180\Z)$.
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 | $4$ | \( 31935 = 3 \cdot 5 \cdot 2129 \) |
| $3$ | split multiplicative | $4$ | \( 21290 = 2 \cdot 5 \cdot 2129 \) |
| $5$ | split multiplicative | $6$ | \( 12774 = 2 \cdot 3 \cdot 2129 \) |
| $7$ | good | $2$ | \( 2129 \) |
| $2129$ | split multiplicative | $2130$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 63870.o
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 |
|---|---|---|---|
| $3$ | 3.3.127740.1 | \(\Z/14\Z\) | not in database |
| $6$ | 6.6.2084398420824000.1 | \(\Z/2\Z \oplus \Z/14\Z\) | 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 | 2129 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | split | split | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | split |
| $\lambda$-invariant(s) | 2 | 2 | 2 | 3 | 1 | 1,1 | 1 | 1 | 1 | 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 | ? |
An entry ? indicates that the invariants have not yet been computed.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.