Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+675574x+8108580\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+675574xz^2+8108580z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+875543877x+375687276822\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(196, 12070)$ | $0.25503608223036862659117698646$ | $\infty$ |
| $(-12, 6)$ | $0$ | $2$ |
Integral points
\( \left(-12, 6\right) \), \( \left(4, 3286\right) \), \( \left(4, -3290\right) \), \( \left(196, 12070\right) \), \( \left(196, -12266\right) \), \( \left(664, 27046\right) \), \( \left(664, -27710\right) \), \( \left(988, 40006\right) \), \( \left(988, -40994\right) \), \( \left(3238, 188506\right) \), \( \left(3238, -191744\right) \), \( \left(6280, 498790\right) \), \( \left(6280, -505070\right) \), \( \left(42238, 8661256\right) \), \( \left(42238, -8703494\right) \)
Invariants
| Conductor: | $N$ | = | \( 5070 \) | = | $2 \cdot 3 \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-19761264961776000000$ | = | $-1 \cdot 2^{10} \cdot 3^{9} \cdot 5^{6} \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{7064514799444439}{4094064000000} \) | = | $2^{-10} \cdot 3^{-9} \cdot 5^{-6} \cdot 13^{-1} \cdot 17^{3} \cdot 11287^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3919809976976821723337239594$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1095063189669138043069802386$ |
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| $abc$ quality: | $Q$ | ≈ | $1.102611128088335$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.081698798067971$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.25503608223036862659117698646$ |
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| Real period: | $\Omega$ | ≈ | $0.12997156027386942001329828371$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 720 $ = $ ( 2 \cdot 5 )\cdot3^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.9665387560508572652813787089 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.966538756 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.129972 \cdot 0.255036 \cdot 720}{2^2} \\ & \approx 5.966538756\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 120960 |
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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 not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $3$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $5$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 1510 & 1551 \end{array}\right),\left(\begin{array}{rr} 110 & 1557 \\ 1107 & 8 \end{array}\right),\left(\begin{array}{rr} 781 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 266 & 11 \\ 1029 & 1540 \end{array}\right),\left(\begin{array}{rr} 1549 & 12 \\ 1548 & 13 \end{array}\right),\left(\begin{array}{rr} 937 & 12 \\ 942 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 75 & 328 \\ 38 & 317 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$9661317120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 507 = 3 \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 338 = 2 \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 507 = 3 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 5070.t
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 390.d4, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.62400.3 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.160398576.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.5922408960000.25 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.90079840281600.43 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.658045440000.15 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.7333751368868611304482312500000000.1 | \(\Z/18\Z\) | 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 | nonsplit | ord | ss | add | ss | ord | ord | ss | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 2 | 2 | 1 | 1 | 1,1 | - | 1,3 | 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 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.